I think this is the nub of the problem. It's a situation familiar to anyone in software engineering: the same routine has been doing two, not actually very similar, jobs, for years on end and separating the two becomes very difficult, as the whole thing means different things to different clients.
For what it's worth, it's not this way in other countries. I know that in Mexico, your church / religious wedding has no legal meaning whatsoever. You then show up in court and have a J.P. do a legally binding ceremony for you.
Same sort of deal in both the US and Canada. The church/religious wedding means absolutely zip. Nada. Zilch. Aside from the religious context, that is. Your marriage will not be recognized by the state at all, unless it is performed by someone legally capable of performing marriages. That person can be a Justice of the Peace, or it can be the minister of some church who has a legal licence to marry people (my father is one of these latter -- he was actually given a wallet card by the province, designating him as legally able to perform weddings). No matter which of these two actually marries you, there is even a particular phrase that must be stated during the ceremony for that marriage to be legally recognized ("by the power vested in me by the province of Ontario").

That said...

I can't actually find a place in the bible where it says that marriage is specifically between a man and a woman. Maybe I'm looking up the wrong terms in the concordance. Anyone care to point out the relevant verse?