Your impression is correct and your own translation is actually better than any of the translations suggested so far. I would only give it a little tweak to make it sound more legalese.

"The plantiff is permitted to avoid enforcement of the judgment by furnishing security in the amount of DM XXX unless the opposing party has furnished security in the same amount."

Background: Judgments by German courts usually contain this clause to ensure that one party does not create a fait accompli by enforcing a judgment before it has become final and unappeallable. If the party, in whose favour judgment was entered, enforces this judgment, the other party may be left without recourse if the judgment is reversed by the court of the next instance. For this reason, the prevailing party is required to furnish security in the amount awarded. That way, the party can enforce the judgment, but the interests of the other party are safeguarded where the original judgment is overturned.

In your case, the wording of this clause would indicate that the plaintiff's claim was dismissed, and the Court's decision as to the costs would be enforceable by the defendant (in Germany, loser pays - their own and the opposing party's cost). In accordance with this clause, the plaintiff may avoid the enforcement (of the order as to costs) by the defendant by furnishing security. However, if the defendant has lodged security in the same amount (of the costs awarded), avoidance of enforcement will not be possible, as the plaintiff could satisfy their rights from the security furnished should the original decision be overturned.