I tend to define classes more by their broad concept than by their abilities. Paladins in D&D don't resemble the Peers or Arthurian Knights in ability (I don't recall Lancelot ever shooting lightning bolts from his sword, or possessing miraculous healing powers), but Devotion Paladins do seem to resemble them in code. If I wanted to play an Arthurian Knight, Fighter and Devotion Paladin would be the first places I'd look. (edit: I forgot about Oath of the Crown because I don't really own anything beyond the core books)

But, D&D has taken old concepts and given them new perspective. Oath of Ancients may be a deviation from what was expected in previous editions, but it takes the concept of an Oath and finds new territory. I can appreciate a bit of a spin, and I've actually just rolled up an Oath of Ancients Paladin just to give it a go. The character still has a slight Arthurian bent, actually; but the source of his power comes somewhat from the service of a benevolent Fey patron. He does what's good and righteous, but in more of an intuitive sense than something so structured as you'd expect from Oath of Devotion, and has a strong love for natural beings of the forest. I like referring to him more as a Green Knight than a Paladin, though.

Oath of Vengeance is another deviation from the norm. I actually don't like it. It feels too Grim-Dark too me. Nevertheless, it grants another alternative in the realm of Guys Who Fight With Swords For Sworn Causes for the people who want it. You can still evoke popular literature and folklore with other class archetypes like this; actually, I think we've got it now so that we can cover more of those areas.

Because I'm not fond of them, and nobody has ever played one at my table, I've never thought of whether or not I'd qualify an Oath of Vengeance Paladin as a Paladin. I'd probably sooner call them Black Knights. But, you can call them whatever you want. For broad classification purposes, Oath of Vengeance Paladin is fine. If in your game you and your players want to say they're Not Real Paladins, then do it. You and your friends should decide how your game deals with that kind of stuff.