It’s called the ‘Pfizer–BioNTech COVID-19 vaccine’ in virtually every language and country (see Wikipedia pages per language). Most people outside of Germany just say Pfizer vaccine because it’s a lot shorter and has some preexisting name recognition.

I haven’t even heard people discuss the ‘where’ when getting the vaccine they just want one of the more effective vaccines (be it Moderna or Pfizer- Johnson and Johnson and AstraZeneca are possibly just as effective but that’s hard to communicate and still being studied).

I think either way is probably fine, both companies are sharing the costs and profits equally.