I get where you are coming from but this just feels like a semantics argument. Just because it’s called marriage in both venues doesn’t mean it isn’t already functionally exactly the way you put it.

It feels like a semantics argument because to a large extent it's a fight about semantics. Most of the people opposed to gay marriage aren't fighting the idea that gay folks should be able to see each other in the hospital, or be covered by each other's insurance, etc - they're fighting the idea that their religious ritual from their homophobic religion should be required to accept gay people and/or that they should be required to accept gay people as being in the same spiritual state as them as a consequence of their ritual. It's why arguments against gay marriage are only extremely rarely about the legal rights and privileges granted by marriage but nearly always about things like "sanctity."

Fully separating the legal and cultural/religious concepts of marriage, including in the language is meant to resolve that by ceding the semantic ground without having to cede any actual rights. You qualify and fill out the paperwork? You're in a civil partnership. Do whatever rituals you want, argue whether or not each other's rituals "count" all you want, everyone gets the same rights legally and the government is not in any way saying your rituals are or are not equivalent to anyone else's.

It's a semantics argument that's worked in the past, to some extent. Before the supreme court made same-sex marriage legal nationally, some states introduced "civil unions" as an alternative to marriage for same sex couples. They were often functionally identical to marriage, but since they weren't called marriages you could get some conservatives to approve them. Not enough to get them passed in red states, but enough to tip the balance in swing states.

They are separate, but the term marriage is often what people fight over.

For example the Catholic Church, the largest sect of Christianity, leader believed that domestic partnerships should be allowed and protected. But the term marriage held a "sanctity" to the church and thus they did not believe in marrying them within the religion.

So if you took the term marriage out of the documents, the issue becomes "I don't believe others should have rights that I have" and even their religion doesnt agree with them. Thus the bullshit of hiding behind freedom of religion to hate/persecute others starts to dissolve.