People should stop using this "noncustodial wallet" phrasing. There is no such thing as a noncustodial wallet. It's either in your wallet, or it's in someone else's wallet. There is no other qualifier that determines the type of wallet. Saying "my noncustodial wallet" is an oxymoron. If you don't have control of the wallet, it's not your bitcoin, though, you could say bitcoin is being held by a legal custodian.
I see this "noncustodial wallet" phrasing being pushed by those who would benefit by blurring the lines between what is cryptographically true with what might legally give them power. When a law is made which states, "You may only hold bitcoin in your noncustodial wallet," it doesn't make sense, but when the everyone is convinced 2+2=5 then it might.
After thinking about it a while, I believe you are right about dropping the terminology. But before we come up with something better, we should at least nail down what the heck they mean with the terms. How about "My wallet" and "The exchange's wallet" ha!
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I think this is better.
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