OP, I hope you know that you shouldn't enter mnemonic recovery words to an online website. There's something fundamentally wrong with that, because your recovery words aren't really only in your sole possession. This is also why I don't like the concept of a non-custodial online web wallet, because both the wallet provider and you must have details that allow you both to derive private keys of such a wallet.
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How does
HD wallet key derivation work across different unrelated wallets? See the graph below
Source: https://raw.githubusercontent.com/EAWF/BTC-Toolbox/3938785f186c76598989cc0aa017ad351483d3b1/Images/KeyDerivationTechnicalOverview.png
It was added to the repository with this commit: https://github.com/EAWF/BTC-Toolbox/commit/3938785f186c76598989cc0aa017ad351483d3b1 But is was removed by the uploader for a slightly insignificant reason, some surviving image copies in Reddit show that it's uploaded by the same user. Link to the commit that deleted it: https://github.com/EAWF/BTC-Toolbox/commit/f75e2b352ec9facc8d2da52b5ec303fb280c3298In layman's terms: there are mathematical and/or cryptographical recipies how to derive a master private key from the mnemonic recovery words. Standardized derivation paths define how to derive child keys from such a master private key (there's more involved, but I spare you the details; it's maybe fun for you to read about it in the book "Mastering Bitcoin" which is available for free
https://github.com/bitcoinbook/bitcoinbook).
All wallets that are compatible with BIP-39 mnemonic recovery words and BIP-32
HD key derivation know the rules and recipies to interchangably derive the same private keys and associated public addresses from the same mnemonic recovery words and derivation path. If they don't do it correctly, they're broken.