There is no single word for "rebirth" in the Pali suttas. There are around a dozen different terms that are translated as "rebirth". "Rebirth" is merely an interpretation of these terms.
Ultimately, "birth" ("jati") refers to the generation of the view or idea of "beings ("satta") and "self" (refer to SN 12.2; SN 23.2; SN 5.10 & SN 22.81). What is loosely translated as "rebirth" refers to the continuation or re-arising of "self" view.
For example, good kamma has done & the mind celebrates: "I have done good". Bad kamma is done & the mind regrets: "I have done bad".
Both good & bad kamma keep the mind trapped in suffering due to egoism.
What is "core" to Buddhism is what the Pali suttas state are "core", namely:
The six elements (AN 3.61)
The six sense spheres. (AN 3.61)
The eighteen applications of mindfulness in relation to the eighteen feelings. (AN 3.61)
The four noble truths. (AN 3.61; SN 56.31)
Dependent origination (which does not explain life after death; MN 28).
Three-characteristics (AN 3.134)
Emptiness (SN 20.7).
MN 38 states the dependent origination taught by the Buddha is visible in the here-&-now.
The Pali scriptures state what the Buddha taught: "leads to disenchantment, to dispassion, to cessation, to calm, to direct knowledge, to self-awakening, to Nibbana".
Where as believing in "rebirth/reincarnation" does not lead to Nibbana & especially leads to passion, which is why MN 117 explains ideas about 'rebirth' are polluted teachings that side with morality only; which is confirmed by MN 60, which states believing in rebirth only leads to morality (the three skilful actions).
AN 3.61 states believing 'past lives cause suffering' is easily refuted & is not the Buddha's teaching.
"Good, monks. You have been guided by me in this Dhamma which is to be seen here & now, timeless, inviting verification, pertinent, to be realized by the observant for themselves. For it has been said, 'This Dhamma is to be seen here & now, timeless, inviting verification, pertinent, to be by the observant for themselves,' and it was in reference to this that it was said.
MN 38
> *These, bhikkhus, are the three sectarian tenets which, when questioned, interrogated, and cross-examined by the wise, and taken to
> their conclusion, will eventuate in non-doing.*
>
> *But, bhikkhus, this **Dhamma taught by me** is unrefuted, undefiled, irreproachable, and uncensured by wise ascetics and brahmins. And what
> is the Dhamma taught by me that is unrefuted, undefiled,
> irreproachable and uncensured by wise ascetics and brahmins?*
>
> *‘These are the **six elements**’: this, bhikkhus, is the Dhamma taught by me that is unrefuted … uncensured by wise ascetics and brahmins.
> ‘These are the **six bases for contact**’ … ‘These are the **eighteen mental
> examinations**’ … ‘These are the **four noble truths**’: this, bhikkhus, is
> the Dhamma taught by me that is unrefuted, undefiled, irreproachable,
> and uncensured by wise ascetics and brahmins.*
>
> *AN 3.61*
'We will listen when discourses that are words of the Tathagata — deep, deep in their meaning, transcendent, connected with emptiness — are being recited. We will lend ear, will set our hearts on knowing them, will regard these teachings as worth grasping & mastering.' That's how you should train yourselves.
SN 20.7
> *All processes are inconstant. All processes are unsatisfactory. ll phenomena are not-self. **The Tathagata** directly awakens to that, breaks
> through to that. Directly awakening & breaking through to that, he
> declares it, **teaches it**, describes it, sets it forth. He reveals it,
> explains it & makes it plain:*
>
> *AN 3.134*
~~
> *Now, the Blessed One has said, "Whoever sees **dependent co-arising** sees the Dhamma; whoever sees the Dhamma sees dependent co-arising."
> And these things — the five clung-to-aggregates — are dependently
> co-arisen. Any desire, embracing, grasping & holding-on to these five
> clung-to-aggregates is the origination of stress. Any subduing of
> desire & passion, any abandoning of desire & passion for these five
> clung-to-aggregates is the cessation of stress.'*
>
> *MN 28*
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.
SN 56.31
> *What I teach now as before, O monks, is suffering and the cessation of suffering.*
>
> *MN 22*