I never had a problem with the various names. People call it the Notes server but I am fine with that. And those who don’t know the difference between Domino and Notes don’t care about XPages either. For them this is all just Notes (to be honest even Lotus Notes). But dropping (brand) names is en vogue at the moment.
I am now going to restart my Notes server
Afaik Domino was the internal project name (like Hannover for the first Eclipse client). There already was a Notes server (and it was called Notes server) but the “Domino” release added all those new web capabilities. At that time they thought it would be better to not just call it Notes Server Next. Would they do it again? I don’t know.
I do like the name but there is no Domino’s pizza in Germany (I am not sure they exist outside of the US at all).
Some years ago Domino’s pizza was even a Notes shop. Maybe they still are. I don’t know.