In theory, I 2256% agree with you. But in practice, “crypto wallet” means different things to different people. I know this from experience (and frustration) struggling to explain it to people who just don’t get it.
To you, to me, and to my bygone friend whom I will call Alice here (although she was so cypherpunk anonymous, he refused to reveal one bit of PII which would partition her anonymity set between Alices and Bobs), “crypto wallet” means exclusive control of your own private keys.
To people who don’t know what “private keys” are, or why they are important, “crypto wallet” means “my Coinbase wallet”—or even nowadays, literally, “my Paypal account”. (Before the autumn of 2020, I used Paypal as a rhetorical symbol in contradistinction to cryptocurrency—whoops!)
What terminology do you suggest using with them? “Self-custody” unambiguously denotes, at least, that you are holding your own money. Is there anything similarly simple and unambiguous, which implies that this is the normal and natural state of affairs?
In my experience, the most frustrating part of this is that people who don’t get it are not necessarily stupid. I have groped in the darkness of their incomprehension—trying to make the little light-bulb go on for some people with PhDs or professional degrees, who certainly have above-average IQs. A secret pseudorandom number that you generate all by yourself, which has exclusive peremptory control of money, must be one of those concepts like pointers and recursion.
Spolsky (2005)
If I may be so brash, it has been my humble experience that there are two things traditionally taught in universities as a part of a computer science curriculum which many people just never really fully comprehend: pointers and recursion.
nullius embraces-and-extends Spolsky:
In the spirit of T. C. May, I will be so brash: It has been my unhumble experience that there are
twothreefour things from an alien world of pure abstractions which many people just never really fully comprehend: pointers, recursion, public-key cryptography, and zero-knowledge proofs.
Aside: Zcash has this problem even worse. I have known people who get public-key cryptography, but who just cannot wrap their heads around a system where you prove in zero knowledge that you validated your own financial transaction.
nullius embraces-and-extends Clarke:
Any sufficiently advanced cryptography is indistinguishable from magic.