I have a set of multiple given names, each of them given by a godfather. In our tradition and law they are equal because treating one of them specially is insulting to the godfathers (which you caught me guilty of).
If given name or surname comes first is regional. Where I come from (Bavaria) traditionally the surname comes first, so even by that metric I wouldn't have a first name.
Source:
Die Reihenfolge der Vornamen stellt keine Rangfolge dar. Nach höchstrichterlicher Rechtsprechung (BGH, Beschluss vom 15. April 1959 – IV ZB 286/58) steht es in Deutschland dem Namensträger frei, zwischen seinen standesamtlich eingetragenen Namen zu wählen.
[..]
Die in der Geburtsurkunde eingetragenen Vornamen dürfen von den Namensträgern im privaten Rechts- und Geschäftsverkehr nach Belieben genutzt werden und sind gleichberechtigt.
> I have a set of multiple given names, each of them given by a godfather. In our tradition and law they are equal because treating one of them specially is insulting to the godfathers (which you caught me guilty of).
I don't see how changing "first" to "given" lessens the insult to those N - 1 godfathers.
And having "multiple first names" is not that rare.
Both on Wikipedia and on Merriam-Webster, "family name" is given as a synonym of "last name" or "surname". In my own language as well, "family name" is simply understood as a part of your legal name with slightly different rules.
Perhaps in genealogy circles your definition applies. But it's certainly not how the word (well, expression?) is used in common language in English, and I bet in other languages either.
That is also why passports have a surname field but not a family name field. Not everyone has a surname but conventions to deal with this vary.
Similarly, passports use the terminology given name because not everyone has been christened and not everyone has a first name - for example I don't.