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).
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
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.
Maybe you're confused because the etymology is not clear to you. I see the term "wedding ring" and I think it denotes that person A and person B are being wed to each other. Joined to each other. The ring symbolizes that joining. In other words, I don't think the ring is named after the event. I think both the ring and the event are named after the verb. In which case it's a very normal name.
Another interesting point is that in some cultures it's quite common for people to get married months or years before they have a wedding. They can go to City Hall and get married, and then later when they have time and money they can schedule their wedding celebration party. If that's the case, then the ring that you use for the first time at your wedding might be reasonably called the "wedding ring".
While I don't see the necessity in wearing a ring in the first place - yet I am a sucker for rings - what always gets me is having a fancy overpriced engagement ring that you're supposed to wear for a short period of time and then a simple wedding ring that you're supposed to wear forever.
You can blame the debeers company for the engagement ring. Just like you can blame Kellogg's for the slogan "breakfast is the most important meal of the day".
Wait, do you know women that stop wearing their engagement ring after they're married? Every married woman I've met (that doesn't have a hands based labor job) wears the engagement and wedding ring. All the women in my family have them soldered together some time after, so they don't independently spin on the finger.