A wearer of an engagement ring receives it when they become engaged, and a wearer of a wedding ring receives it when they wed. Seems pretty consistent to me.
Marriage is more analogous to a birthday. (A personal change in status)
Wedding is more analogous to a birthday party (i.e. the event celebrating the change in status).
As you pointed out in your logic, the birthday gift isn't really about the birthday party, just like the ring doesn't commemorate the wedding celebration, it commemorates your new marital status.
Unless of course you are the kind of person that is so focused on the wedding celebration that you forget the reason why you are celebrating to begin with (spoiler: you are making a commitment and entering a new life stage).
A wedding ring shows that you are wed. Just like an engagement ring shows that your engaged.
A wedding is an event, being wed is a change in status. The logic is sound. Confusion only enters the mix, because 'being wed' is less common as a phrase than 'being married'.
Your logic makes sense. To OP's point, though, you wear an engagement ring to show that you are engaged; a wedding ring to display you are married/wed. The argument for it being called when you receive it is weakened by the fact that most people remove their rings when an engagement is broken, or they get divorced. Or, they move the ring to a different finger, at which point it's no longer an engagement or wedding ring, right? It's just a ring.
If the rings were named after the event of reception, they'd still be called wedding and engagement rings even after a broken relationship. They're "was" rings; ex-wedding-rings. No longer engagement rings.
So the more I think about it, the more I'm with OP - the rings represent a state, and so wedding rings should be called "marriage" rings to represent the state of being engaged/married, rather than the singular event of the giving.
I get that, but also, it's always seemed like the purpose of the ring is to signify the state of being, so engagement ring to show the state of being engaged, wedding ring shows the state of being married, or wed, I guess it works both ways