We don’t need to consult many buddhist texts to read that according to them death is not the end of it all. Were it so a knife would then already have been a far more easier tool. Rebirth and reincarnation don’t have this problem but share the idea that a certain something is carried over, or remains, from one life to another and that is there the problem.
Eternalism (sassatavādā) and nihilism (ucchedavādā) are the two worldly views on existence. Eternalism opposes nihilism and must first hold time as never ending (the here forever part). The breaking up, disintegration, perishing, of the existence-through-time is what nihilism is (the here not forever part). And both rebirth and reincarnation are based on these misconceptions. And though used as argument against nihilism, existence-through-time until-no-more is really just a delayed nihilism which is how "enlightenment" can be seen as not appealing. Now for something you never ever have experienced, heard of, etc., you have no name. So if you do have a name it is for a certain experience, but when this is misunderstood the name includes the misunderstanding. Things can be remembered (SN22.79), thought out, perceived, yet misunderstood. And so do rebirth and reincarnation not just indicate life after dead, they must also include a certain how. And depending on this how these views can be seen different or as synonyms, but what these outsiders’ views did not and do not include is dependent-co-arising.
If rebirth and reincarnation were to be redefined to imply dependent-co-arising, a buddhist version of rebirth or reincarnation? why such horrible construct? it would still solve nothing. Rooted in wrong view they don’t disentangle anything but build further on top what is not understood; it added another concept to the confusion thus requiring more explanation not less. Just as with eating; it doesn’t matter how often you eat it stays eating (eating is eating) you wouldn’t then suddenly re-eat which would then also require more of an explanation not less: 'Is it like what cows do?' So this redefinition engraves and masquerades the already underlying wrong views which now forms support for pernicious explanations like dependent-co-arsing with lifetimes. But dependent-co-arising explained as existence through time, lifetimes, rebirth, stays to be sassatavādā & ucchedavādā. Instead of rebranding rebirth and reincarnation it is now dependent-co-arising that has become subjected to this car cloning as the world must keep faring towards opposite sides. So what happens is that this "dependent-co-arising" now affirms rebirth and reincarnation as it is molded into them; it has become the new how for these views and justifies them. The middle has been trown out, the jargon kept.
With rebirth and reincarnation being are seens as reborn or reincarnated, in any case beings are born. Both views must at least acknowledge birth. And it is precisely birth which is not being understood. The texts themselves talk f.i. about birth, next birth, a following becoming; translations as rebirth and reincarnation are just translational liberties but unlike them birth is not just a view. Whether one understands it or not, birth already includes the possibility for a next birth (birth is birth) for it is dependently-co-arisen. The usage of next birth, future birth, further becoming, is already correct and enough to explain that death may no need to be the end of it all. There is no need to seek shelter in wrong views, two wrongs don't make a right.
The world fares towards opposite sides (eternalism, nihilism). It was the Tathāgata who taught by the middle, and the middle simply can not be grouped with either side.