"A Spanish language name is simply called that: whereas a Spanish name is the name of a Spanish person."
Not in onomastics works, which are linguistic matter. In Spain, a lot of people use names in English and these are not Spanish names, are simply English names (names in English) used in Spain/by Spanish people: Kevin, Jessica, Vanessa... (Probably with time they could be assimilated to Spanish and considered Spanish names [from English origin], but this is a long way.)
And Basque names are used not only by Basques (in North and in South) or Spanish, but in Idaho... are they Basque names or are they American names? A Basque name is a name in Basque, used in Bilbao, in Barcelona or in Bloise.
"English national adjectives simply do not work that way."
If you check reliable sources of Welsh names (as Gruffudd's works or Iain O'hAnnaid's website) or of Irish names (as Ó Corráin and Maguire's work) you can verify that English national adjectives in names work in that way. In onomastics and in linguistics at least.
"For example, if I told you a name was Bengali, I would mean that it is a name commonly used by people who you and I called Bengali, and not much by others. Note that the class referred to are not necessarily people who call themselves Bengali, except in so far as that coincides with who we (you and I) call Bengali."
Then, the Indian use is perhaps the exception in onomastics because its reality in languages, religions, cultures... I don't know, I believe you because you know the Indian reality, so if you say "Bengali name is a name used in by Bengali people", I can think that this is not linguistically/onomastically accurated but I accept that in references to India, the adjective refers to geographical origin and not to linguistic adscription.
But if you say Welsh name, Basque name, Sardinian name, Breton name, Occitan name, Finnish name, Nahuatl name, Armenian name, Kurdish name... the adjective indicates the origin language of the name and it is not a reference to the passport of person who is named this.