using the phrase: 12345678901234567890 (brain wallet)
This phrase make a private and public key that apparently have been used for real before.
Private key:
5Jf6mGTeCg37iZST1ukHXacwyuhjL3ABvEcpgwCRQX9y9fLGLbD (thats the same key the OP found/created)
Public key>
15eUHuCMKGekXmfH2B5awPWnwY3qVe3Nz1
Yeah I think you got it right.
This basically just means that it is insecure to use bad randomness to generate your Keys. But that's an obvious thing and does not only apply to Bitcoin but to crypto in general.