That is true, a judgment is vacated as part of a motion to set aside a default. The judge will vacate the default and any recorded judgment. There is no more judgment, but even then, depending upon the Jurisdiction, the record of the now vacant Judgement may still be found in the record. Marked as vacated and rendered unenforceable, but there.
If the judgment is paid and satisfied, the judgment is eliminated. You are right, it is part of the public record, it too remains, marked as satisfied.
You can't vacate a judgment just because it was paid, not at least in NY. Even if you did, it would be of little effect, as again they do not erase those things. Even with a 522 motion in Bankruptcy, even though the lien is unenforceable by discharge; render avoided by the 522 motion and the Court will order it to be "vacated and expunged" the judgment itself remains on the record (dead as a doornail), again marked according, but it can be found. Again at least in NY.
If after paying a judgment you wanted to "undo it" and get your money back and moved to vacate the judgment. The Court would likely refuse,the judgment having been paid and satisfied it is no more and that Judge/Court technically has lost juirsdiction to do anything with it. You would have to file an action under some theory to recover against the Creditor.
