Can I test that I've correctly recorded the seed by "restoring" my wallet in a desktop wallet like Electrum by using the seed given to me by my Nano S? It would mirror my hardware wallet, correct?
Is this safe / right way ?
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If you do this correctly, it would mirror your wallet, yes. BUT!
Doing so is probably not a good idea, because you'll have to type your seed phrase into a desktop computer that may be compromised. The point of a hardware wallet is keeping your keys locked away in a device which never touches the internet, testing your backup seed phrase on an internet-facing device entirely undermines this.
Instead, it is best to test your seed phrase using the hardware wallet itself (or, a second one).
Initialize the hardware wallet, write down your seed phrase
Open the wallet software on your PC, copy the receiving address your wallet shows into a text file
Wipe/reset your wallet, and re-initialize it using the seed phrase
Once done, open the wallet software on your PC, and check to see if the receiving address is the same as the one you copied to your text file. If it is, then you successfully recreated the same wallet.
If you've already sent funds to your hardware wallet, then it may be worth removing the funds from the hardware wallet before doing this test (or, just do the recovery test on a second hardware wallet). You wouldn't want to wipe the wallet with funds on it, before you're comfortable with the recovery process.