There is an term 'wedding', when you put your signature and it becomes official. But what is the right term for a religious act when two swear to be faithful, to love each other e t.c. in front of the God and a priest in church? Is there such term in English speaking countries? I mean, in some countries young people prefer to "marry" only in church and not to sign any documents officially. I heard about such terms as 'crowning' or even 'church wedding' but I'm not sure it is correct to use any of them.
It might depend not only on regional customs but religious customs as well. The term that I am familiar with in AmE is church wedding as you suggested:
a wedding ceremony performed in a church and having a religious rather than civil content
Wedding by itself can be somewhat broad. It does not necessarily mean signing a marriage license and becoming "official". It really depends on the context, taking regional, religious, and personal factors into consideration.
As for the "official" one, as far as I know, that's a civil marriage or a civil wedding.