The mnemonic recovery words, most commonly as defined by BIP-39, are a human readable backup representation of the initial secret entropy from which all else is derived for a HD wallet. This initial secret, be it commonly 128 or 256 random bits long, is what defines your personal HD wallet. DO NOT store your mnemonic recovery words unencrypted on any online device. Keep them offline and preferably analog on paper and additionally for hazard protection stamped in metal.

You're not wrong how you express it, but beginners should understand the basics more clearly in my opinion. Maybe this graph helps a little bit (it's not easy to digest if you aren't familiar with some of the technical details though):


Found the 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 it 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/f75e2b352ec9facc8d2da52b5ec303fb280c3298


The key points to take away are that software or hardware wallets that commonly understand the BIP-39 standard can reproduce a HD wallet from those mnemonic recovery words. Most wallets also follow pretty standard derivation paths to derive address types and private keys, public keys and public addresses accordingly.