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But if that's so, one thing that still confuses me is with https://walletgenerator.net > Paper Wallet, you can create paper wallets with your private key. But if you reapply the same key you get different addresses. What's up with that?
A standard (BIP32, many math) has been specified in the meantime to derive multiple keys from a single "seed" deterministically, so that with the single piece of random data, you can recalculate all the individual private keys as long as you specify the same parameters. Most modern wallets, including those created with Dogecoin Core v1.14, implement this. Most often, the seed is encoded as a set of words.