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.
For something you never ever have experienced, seen, heard of, etc. you have no name. If it has a name it is for a certain experience. But if this is misunderstood that name includes the misunderstanding. Things can be remembered (SN 22.79), thought out, perceived, yet misunderstood. And rebirth and reincarnation do not only talk about life and dead, they also include a certain what and how. And depending on this these views can be seen as different or as synonyms. But what these outsiders’ views can not include is dependent-co-arising. If rebirth and reincarnation were to be redefined to imply dependent-co-arising it wouldn't disentangle anything as they build further on top what was not understood. It added another concept to the confusion, now 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 now require more explanation not less 'was it about vomit? or what cows do?' So this redefinition engraves and masquerades the already underlying wrong view.
Dependent-co-arising explained as existence through time, x-lifetimes, 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 can only go by both ends. So what happens is that this "dependent-co-arising" now affirms rebirth and reincarnation as it is molded into them. It becomes the new how for these views and justifies them. The middle has been thrown out, the jargon is kept. With rebirth and reincarnation beings are viewed as reborn or reincarnated, yet both views must thus at least acknowledge birth. And it is precisely birth which is not being understood. Where rebirth and reincarnation are used, the Pāḷi texts talk about birth, next birth, a following becoming, etc. Birth is not just a view and it already includes the possibility for a next birth (birth is birth) for it is dependently-co-arisen. The terms rebirth and reincarnation don't add value but will lead people astray as they affirm the wrong views on existence. There is no reason to seek shelter in these wrong views, two wrongs don't make a right.
The world is having to go by the both ends (eternalism, nihilism). It was the Tathāgata who taught by the middle, and the middle simply can not be grouped with either end.