Will this be different from the Bitcoin keys ? After all, nostr is a hot ‘wallet’, and there are often clients (even web client) that encourage users to paste the private key. If the user pastes the private key derived from Bitcoin, the whole wallet will be exposed.
I totally forgot to mention in my talks at #[0] that you can derive a Nostr key from a BIP-39 mnemonic backup phrase.
* You (or more likely your Nostr client app) generates a new mnemonic backup.
* Write it down and protect it just like a Bitcoin seed.
* Client app derives your Nostr key from that seed.
#[1], #[2], #[3], #[4] for app-generated Nostr keys, it'd be nice to go through NIP-06 derivation (`m/44'/1237'/0'/0/0`) and provide the mnemonic as their backup.
Eventually when we get keys out of browsers, we'll want more robust backup solutions. May as well piggy back on what's already out there (e.g. steel plates) for Bitcoin.
Can generate BIP-39 -> NIP-06 keys via https://nostrtool.com

Discussion
You are correct.
One should NOT use their Nostr BIP-39/NIP-06 mnemonic phrase as a Bitcoin wallet, even though they are perfectly compatible.
Nor should someone use their existing Bitcoin BIP-39 mnemonic to derive their NIP-06 Nostr key.