I suspect the Buddha used the term "rebirth" in a sense different from its traditional meaning. Probably he meant that our sense of separate self is repeatedly being born, decaying, dying and being reborn until we attain Nirvana. In this sense, rebirth happens only in our present life and not after our physical death. Is there any evidence in the early canon to support that the Buddha used the term "rebirth" in the above sense?
I suspect the Buddha used the term "rebirth"
The entire premise of your question is faulty, unfortunately. The Buddha never, afaik, used a term that could be translated as "rebirth". In fact, the idea of anything being reborn goes against orthodox early Buddhist teachings. Throughout the Buddha's teachings, it is made clear that at the breakup of the body there is birth, not rebirth - as in birth of new things, not the return of anything old.
Probably he meant that our sense of separate self is repeatedly being born, decaying, dying and being reborn until we attain Nirvana.
Since the Buddha never taught the idea of self and denounced any view of self as leading to suffering (MN 22), this is highly unlikely to be correct.
In this sense, rebirth happens only in our present life and not after our physical death.
To repeat, according to early Buddhism, rebirth never happens. In this life, there is what is called khaṇika-maraṇa - death of a moment. Each momentary experience is born and dies, never to arise again. At the moment of conceptual death (sammuti-maraṇa), this process of momentary birth and death continues unimpeded unless one has experienced "death by cutting off" (samuccheda-maraṇa) - i.e. of the defilements (q.v. Vism VIII.167) - in which case there is no further arising.
To put it succinctly, physical death isn't even real according to early Buddhism; it is a concept based on the artificial construct of a "being" who is "born".
Is there any evidence in the early canon to support that the Buddha used the term "rebirth" in the above sense?
On searching the Pali cannon for the words 'death' and 'reborn' there are many suttas which include the phrase,
... with the breakup of the body, after death, is reborn ...
In this Translator's Introduction, Thanissaro Bhikkhu wrote,
This sutta teaches how to understand the relationship of consciousness to rebirth in a way that helps put an end to rebirth.
Although the Buddha never used any word corresponding to "rebirth" in his teachings, he did describe birth as a process following on death again and again as long as the appropriate conditions are present. In other words, even though he didn't use the word "rebirth," his teachings on birth are teachings on repeated birth: how it happens, how it inherently involves suffering and stress, and how it can be brought to an end.
The idea that death can be followed by birth was not universally accepted in India in the Buddha's time. As DN 2 and MN 101 show, some prominent contemplative schools actively rejected the idea of rebirth while others affirmed it. Thus when the Buddha taught rebirth, he wasn't simply following an unexamined cultural assumption. He was consciously taking a stand on one of the controversial issues of his time. However, his explanation of rebirth differed from other schools on both sides of the issue in that he avoided the question of whether or not there's a "what" that gets reborn, or if there is a "what," what it is (SN 12.12; SN 12.35). He also discouraged such speculations as, "If I take rebirth, what was I in the past, and what will I be in the future?" (MN 2)
He also writes the following, which agrees with what you asked in the question (not rebirth-during-this-life instead of rebirth-after-death, but at least rebirth-during-this-life as well as rebirth-after-death),
However, a being — in the Buddha's sense of the term — not only takes birth after the death of the body, it can also take birth, die, and be reborn many times in the course of a day — as attachment develops for one desire, ends, and then develops for another desire. This is why the processes leading to rebirth after death can be observed and redirected in the present moment during life. This is why the ability to understand and observe the processes of dependent co-arising is so important in putting an end to rebirth on all its many levels.
I offer this article, "A Secular Evaluation of Rebirth", as a possible answer to your question.
It starts with,
Insofar as we know anything about his dhamma, we know that the Buddha taught literal rebirth.
After looking at evidence gathered by Ian Stevenson, considering whether memories are trustworthy, and whether we have any physical theories for rebirth, it concludes,
It is for reasons such as these that any contemporary, scientifically informed Buddhist practice should reject belief in rebirth and its associated kammic causation. The Path is rich enough without them. And while we can make good use of kamma and rebirth as metaphor for our moment-to-moment lived experience of change, or of skillful and unskillful action, we simply cannot make any more of it and expect to end up with a system which is compatible with our best understanding of the way the world works.
In general, I think the Pali canon does not agree with your interpretation, though the article makes clear that there are Buddhists who do (I am one of them). You might look to "Secular" Buddhism for more about this.
First I will offer some background before I say anything about what the Buddha said or didn't about rebirth.
Logically speaking, both view points are true since consciousness is not affected at the time of physical death or rebirth.
You mention in the comments that you are not comfortable with the traditional idea of rebirth, which is okay, take what you can verify and trust. If tomorrow you change your mind because of new insights, that's okay. If not, that's okay too. You don't have to agree with others, be independent using your logical faculties as your guide.
My submission is that in 2500 years the numerous brilliant minds in Buddhism and other Indian traditions were not all so dense as to not be aware of material reality, and there were certainly some schools (Lokayatika) who were only concerned with impermanence in the material reality.
In fact there is the Lokayatika Sutta where the Buddha specifically confronts the position of material reality being the only truth.
Heraclitus, a near contemporary of the Buddha has a pithy saying that "You could not step twice into the same river." meaning the body and mind is ever changing. This is often quoted by modern dharma teachers in the West because it perfectly reflects the Buddhist teaching on emptiness, and impermanence.
Here is what the Dhammapada says on impermanence in the present moment -
Sabbe sankhara anicca" ti yada pannaya passati atha nibbindati dukkhe esa maggo visuddhiya.
Dhammapada Verse 277: "All conditioned phenomena are impermanent"; when one sees this with Insight-wisdom, one becomes weary of dukkha (i.e., the khandhas). This is the Path to Purity.
Certainly traditional rebirth is heavily referenced in the earliest sutras of the Buddha.
There are many references to rebirth in the early Buddhist scriptures. These are some of the more important; Mahakammavibhanga Sutta (Majjhima Nikaya 136); Upali Sutta (Majjhima Nikaya 56); Kukkuravatika Sutta (Majjhima Nikaya 57); Moliyasivaka Sutta (Samyutta Nikaya 36.21); Sankha Sutta (Samyutta Nikaya 42.8).
See more: Wikipedia-Rebirth(Buddhism)
The specifics of the concept of rebirth however was left sufficiently vague by the Buddha, since he was first and foremost concerned with liberation from suffering, and not in solving philosophical debates of the intellectual kind.
This has led to several schools of thought regarding a persistent self and nuanced details of rebirth. However, there were no Buddhist schools of thought denying rebirths, which may be instructive.
The Mahāyāna Mahāparinirvāṅa Sūtra, especially influential in East Asian Buddhist thought, goes so far as to speak of it as our true self (ātman). Its precise metaphysical and ontolo-gical status is, however, open to interpretation in the terms of different Mahāyāna philosophical schools; for the Madhyamikas it must be empty of its own existence like everything else; for the Yogacarins, following the Laṅkāvatāra, it can be identified with store consciousness, as the receptacle of the seeds of awakening.
Source: The Foundations of Buddhism, Gethin, 1998, page 52
I'll finally leave you with this,
Once the Blessed One was staying at Kosambi in the simsapa1 forest. Then, picking up a few simsapa leaves with his hand, he asked the monks, "What do you think, monks: Which are more numerous, the few simsapa leaves in my hand or those overhead in the simsapa forest?"
"The leaves in the hand of the Blessed One are few in number, lord. Those overhead in the simsapa forest are more numerous."
"In the same way, monks, those things that I have known with direct knowledge but have not taught are far more numerous [than what I have taught]. And why haven't I taught them? Because they are not connected with the goal, do not relate to the rudiments of the holy life, and do not lead to disenchantment, to dispassion, to cessation, to calm, to direct knowledge, to self-awakening, to Unbinding. That is why I have not taught them.
"And what have I taught? 'This is stress... This is the origination of stress... This is the cessation of stress... This is the path of practice leading to the cessation of stress': This is what I have taught. And why have I taught these things? Because they are connected with the goal, relate to the rudiments of the holy life, and lead to disenchantment, to dispassion, to cessation, to calm, to direct knowledge, to self-awakening, to Unbinding. This is why I have taught them.
"Therefore your duty is the contemplation, 'This is stress... This is the origination of stress... This is the cessation of stress.' Your duty is the contemplation, 'This is the path of practice leading to the cessation of stress.'"
The interpretation you offer is not in line with the Buddha's teachings as I understand them. Other, western Buddhists have suggested similar interpretations, however, literal reincarnation is a repeated theme throughout the suttas. Here are a few reasons that I think the non-literal or 'this lifetime' interpretation of rebirth is not tenable.
(1) When discussing rebirth, as ChrisW pointed out, the phrasing that thematically appears makes it clear that it is meant to be literal, as in the Saleyyaka Sutta:
"Householders, it is by reason of conduct not in accordance with the Dhamma, by reason of unrighteous conduct, that beings here on the dissolution of the body, after death (emphasis mine), reappear in states of deprivation, in an unhappy destination, in perdition, even in hell. It is by reason of conduct in accordance with the Dhamma, by reason of righteous conduct, that some beings here on the dissolution of the body, after death, reappear in a happy destination, even in the heavenly world."
The same phrasing appears elsewhere, e.g., in the Maha Kammavibhanga Sutta.
(2) The Buddha occasionally declares the destination of deceased persons; i.e., he says where they have been reborn (or that they will not be reborn).
(3) The Buddhist cosmology explicitly declares the existence of various hell and heaven realms and beings that populate them. These are possible destinations after death for a human being, as are rebirth as an animal or a hungry ghost.
(4) The Buddha distinguishes between the fruits of kamma in the 'here and now' and those in future births. That the Buddha is talking about reincarnation is made perfectly clear by this conversation with his disciple, General Siha, in the Siha Sutta:
When this was said, General Siha said to the Blessed One: "As for the four fruits of generosity visible in the here & now that have been pointed out by the Blessed One, it's not the case that I go by conviction in the Blessed One with regard to them. I know them, too. I am one who gives, a master of giving, dear & charming to people at large. I am one who gives, a master of giving; good people, people of integrity, admire me. I am one who gives, a master of giving, and my fine reputation is spread far & wide: 'Siha is generous, a doer, a supporter of the Sangha.' I am one who gives, a master of giving, and when I approach any assembly of people — noble warriors, brahmans, householders, or contemplatives — I do so confidently & without embarrassment. (Context note: The Buddha has just finished outlining the fruits of generosity in this life; they are those which the general says he enjoys, i.e., one is dear and charming, admired, of good reputation, and approaches assemblies confidently).
"But when the Blessed One says to me, 'At the break-up of the body, after death, one who gives, who is a master of giving, reappears in a good destination, the heavenly world,' that I do not know. That is where I go by conviction in the Blessed One."
"So it is, Siha. So it is. At the break-up of the body, after death, one who gives, who is a master of giving, reappears in a good destination, the heavenly world."
Again, the general directly knows how generosity leads to good results and certain consequences within the present life, but needs to rely upon the Buddha in faith for knowledge of what happens after death.
I think there are more reasons but this should be enough to establish that the Buddha of the suttas was talking about literal rebirth.
Finally, in the Kalama Sutta, the Buddha declares that if one undertakes right actions, they will benefit in the here and now and can also be assured that (although they doubt the doctrine of reincarnation) if there is rebirth that is causally connected to one's deeds, then one will be born in a good destination.
Further reading: For what I think is an interesting discussion of this and other issues of western Buddhist modernism, see B. Alan Wallace's critique of Stephen Batchelor's views on Buddhism: Distorted visions of Buddhism: Agnostic and atheist and Batchelor's reply: An open letter to B. Alan Wallace