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It's called a wedding ring, but surely it should be called a marriage ring
(startrek.website)
A "Showerthought" is a simple term used to describe the thoughts that pop into your head while you're doing everyday things like taking a shower, driving, or just daydreaming. The best ones are thoughts that many people can relate to and they find something funny or interesting in regular stuff.
Your logic is sound, but backwards.
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 think OP is on to something.
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'.