The simplification I am presenting here is more accurate than saying a Bitcoin wallet “receives and stores bitcoin,” which never, ever happens and never has happened. It is also wrong to characterize a Bitcoin wallet as “unhosted” if it can sign a message on command of a user without reference to anyone else. There are no “wallets” in Bitcoin at all. It’s just another analogy.

Bitcoin is a database. It is not a “payment network” nor is value “sent” over it at all. There are no “wallets” either. Signed messages are what are sent to the network for inclusion in the public database. It is a database used to keep a record of who controls which outputs. It is not — and never has been — money in the conventional sense. Just because people use this database as money doesn’t mean that bitcoin is money. Just because people use the word “wallet” does not mean that there are actual “Bitcoin wallets” that hold bitcoin the way a leather wallet holds cash.

Using the word “wallet” for the sake of user experience is a convention to help make the primary function of tools understandable for users.