"just like an unqualified use of the term Spanish name somewhat implies that it is not restricted to the Basque population, and that it is probably not used in France"
This is a very bad example.
Spanish name simply means "name in Spanish", Basque name simply means "name in Basque". The "adjectivation" regarding a name should be refered to language, not to political administrations. If pointing the geopolitical use is important, the label should be "name used in...".
Basque names are not only used in Basque Country under Spanish jurisdiction, but in Basque Country under French jurisdiction. And, regarding the political feelings among Basques and Catalans, to call "Spanish name" a Basque name (or a Catalan name) can be offensive.
I don't know the political/language/religious situation and the feelings in India, but I know very well the political/language situation in Catalonia, Basque Country and Spain.