Regarding wife's name in marriage certificate

[Sukhjeevan]

Hello,
My wife has her name "namea" ( before marriage ) in marriage certificate but she has new name "nameb" ( after marriage name changed both firstname and lastname ) in passport. We also have newspaper cutting photocopy mentioning her name change from "namea" to "nameb". While going for for H4 visa stamping should we carry newspaper cutting as a proof of name change OR instead of that we have to change her name in marriage certificate to "nameb" which is in passport?

Does marriage certificate and marriage registration certificate is same terminology used for marriage certificate.

Thanks