Wallet
Software or a device that stores private keys and uses them to sign transactions; it does not hold the assets themselves.
Balances live on the blockchain, and the wallet's job is to hold the keys that authorize changes to them, build transactions, estimate fees, and read the chain through a node or a hosted service. A hot wallet keeps keys on an internet-connected device for convenience, while a cold or hardware wallet keeps them on a device that signs internally and never exposes the key. A custodial wallet is different in kind: the provider holds the keys and the user holds a claim against that provider, which is a relationship governed by contract and law rather than by cryptography. Because most self-custody wallets derive every key from one seed phrase, backing up that phrase and nothing else restores the whole wallet.
In practice
Moving from one wallet application to another is done by importing the same seed phrase, because the addresses are derived from the phrase rather than stored by the app.
The common misunderstanding
A wallet does not contain coins; it contains keys, so switching wallets moves key management while the assets stay recorded on the network.