In spoken English it is not necessary to specify 'first' name. We using 'name' in this context the fact you are referring to the first name is implied.
Given name should also be accepted, in fact that's how it is on my passport: Given names/Prénoms. Reported.
Also in many English documents Christian name is used
I agree completely. In European cultures given names usually come before surnames, but there are other cultures. I just heard someone refer to Deng Xiaoping as Chairman Xiaoping.
For people learning French: prénom/prenume. Same meaning.