Michael Saylor sums up the game theory pretty well
https://noagendatube.com/videos/watch/cd788d1f-39f5-4c52-9357-85d526269736
A wallet that someone else runs on your behalf may or may not have anything to do with actual bitcoin, but rests instead on someone else promising to handle your funds and transactions on your behalf. "Don't trust" is the first part of "Don't trust, verify." If your bitcoin is under someone else's personal control, you don't have bitcoin at all -- you have someone's promise. The eventual future value of such promises tends toward $0.
A "Savings Account" or "Cold Storage" can build only unsigned transactions. One or more hardware wallets are required to sign these transactions for broadcast to the network.
Often, people have several wallets: A "Checking Account" or "Hot Wallet" on an internet connected device is in full control of funds as the software knows the required seed+passphrase. The hot wallet can build and sign transactions by itself, without requiring a separate device. Access to this device gives full access to your funds, so this is a model for small balances only.
Seeds and Passphrases are a human-friendlier way to store the public/private keypairs that software wallets and signing devices use to sign transactions. Long binary or hexadecimal strings are encoded as words and/or numbers. Easier to write down on paper or elsewhere. Just remember: anyone who finds your seed phrase and passphrase can import them into their own wallet and then spend all your bitcoin.
A Signing Device, or Hardware Wallet (confusing name!) can sign the transactions generated by your wallet. Never plugging it into a computer or putting on the network makes for a much safer way to store your keys. Some hardware wallets (Coldcard) can also generate receiving addresses on their own, so you can verify for yourself that the wallet software and singning device are generating the same receiving addresses.
A Wallet can track your personal balances and build transactions that send and receive bitcoin. If you give the wallet your public and private keys, it can build and sign transactions, spending your funds. As this software runs on a computer or phone, it is usually preferable to only provide your wallet with your public keys. Such a wallet can build transactions for your signing device to sign, but can't sign them itself.
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