This is a very good question.
Person/ being/ self
A person (puggala) to me looks like the story of someone, as defined in SN 22.22.
A being (satta) is defined as the assemblage of the five aggregates (SN 5.10) and also when there is a clinging to the five aggregates (SN 32.2).
A self (atta) is just an ephemeral mental idea (Snp 4.14) that arises when the five aggregates work together (SN 35.205).
According to the Abhidhamma, all of the above arise and cease at every mind-moment. From one mind-moment to another mind-moment, it's connected by chains of dependent conditioned processes, an example of which is given in this answer. The five aggregates are dependently arisen (MN 38).
Also, a being that arises in one mind-moment (or moment of the dependent arising of five aggregates), and ceases, then arises again in the next mind-moment (or moment of the dependent arising of five aggregates), is sustained or connected by craving (SN 44.9). This is similar to Heraclitus' river and is very well explained in this answer.
This is birth (jati), not rebirth, as Ven. Yuttadhammo had explained in this answer.
It's important to note that a being (satta) is just a view (ditthi).
Why now do you assume 'a being'?
“Kiṁ nu sattoti paccesi,
Mara, have you grasped a view?
māra diṭṭhigataṁ nu te;
What about karma and self? That's explained in SN 12.17.
What about from one "life time" to another "life time"?
There is a sutta about our body and mind being old kamma (SN 35.146). It appears to me that this comes from our surroundings - our parents, family, friends, school, books, media etc. The sutta says "old kamma" but it doesn't say whose old kamma. It just says "old kamma". And that corresponds to the candle flame transmission analogy of Milindapanha.
The Abhidhamma does talk about "rebirth-linking consciousness" and some people connect this to the "consciousness descending into the womb" (DN 15).
But the latter ("consciousness descending into the womb") simply denotes the appearance of consciousness in a name-and-form (mind-body) and states that consciousness and name-and-form (mind-body) as being dependent on each other (as discussed in SN 12.67). In other words, you must have a body formed in the womb, before you can have consciousness and mind-body. And all this arising from ignorance. Again, this is birth (jati), not rebirth, as Ven. Yuttadhammo had explained in this answer.
The idea of "rebirth-linking consciousness" doesn't find much support in the suttas. The right view of dependently arisen consciousness is explained in MN 38, and that leaves no room for "rebirth-linking consciousness". Another idea called "consciousness without surface" was also debunked in this question, and discovered to be a mistranslation of "vinnanam" in that context, and meant to be a description of Nibbana. So any right view of consciousness must only come from the definition of MN 38 and also "The All" (SN 35.23).
Of course, if "rebirth-linking consciousness" simply refers to the mind/ intellect component of old kamma (SN 35.146) that is inherited from our surroundings and also corresponding to Milindapanha's candle flame transmission analogy, then that makes sense. But if it is used to imply the transmission of a fixed consciousness, then that's the Hindu idea of Atman and is not Buddhism. It's debunked in MN 38.
The view of rebirth (upapatti) of a specific self/ person is of course, used as skillful means (upaya) as explained in AN 5.57 and this answer, to remove the habit of misconduct, cultivate virtue and generate the path to liberation.
Buddha's past lives
What about MN 81?
What is special about sammasambuddhas compared to arahants and paccekabuddhas? They have an extraordinary ability to teach that others do not have. One of these abilities is skillful means (upaya).
We should look at skillful means as medicine prescribed by the Buddha, as the unsurpassed doctor (Iti 100), for some specific illness. Yes, it's the same kind of skillful means described by the Parable of the Burning House in Mahayana's Lotus Sutra.
For example, in DN 16, he recommended stupas as a way to inspire the mind, and stated in AN 7.52, that the highest purpose of charity is to adorn/ brighten the mind. So, charity and stupas here are used as medicine for the mind. There's more in AN 5.57.
Similarly, in MN 81, the Buddha used the story of his past existence as Jotipala, as skillful means (upaya), to inspire the mind of the listener. The listener can see from here, that virtues can lead to liberation, and he can be inspired by it. This is of course, only my opinion and speculation, and may not be endorsed by the official traditional commentaries of the Theravada tradition.
In MN 81, the Buddha stated "I myself was the student Jotipāla at that time" but remember that he qualified "I" used by the fully enlightened ones in the sutta below:
“Someone who has given up conceit has no ties, the ties of conceit are
all cleared away. Though that clever person has transcended identity,
they’d still say, ‘I speak’,
and also ‘they speak to me’. Skillful, understanding the world’s
conventions, they’d use these terms as no more than expressions.”
So, is Gautama Buddha the reborn person of Jotipala? Or not? Or both? Or neither?
The answer to this question is very well explained by Yamaka Sutta (SN 22.85) and Vajira Sutta (SN 5.10).