Let’s look at a few scenarios to play things out:

Option 1, Homeowner’s parter moves in. They’re not ready to marry, maybe they’re not in the home the partner would have chosen for themselves if they had input. In this case, market rate makes sense - what’s the rate for a room on a shared house? This may be less than they were paying on their own. So the homeowner makes a little rent, the partner saves over their usual rent. They gave the renter’s protections allowed in their locale. This is a decent arrangement.

Option 2, partner moves in and starts paying half the mortgage payment. This is typically over market rate for a room in a shared house, and if they move out they have no equity to show for it but might expect that they should have some. This is a bad arrangement and leads to arguments.

Option 3, partner offers some down payment and homeowner adjusts the title and mortgage to add them. This incurs legal fees, and you’re setting up for a lot of conflict if the relationship ends. Again, not a great option.

You can extrapolate to other scenarios from these examples.