Yes, these two words are not referring to the same thing at all. One is describing an action (concluding a written agreement) and the other is almost certainly describing a document that one party's lawyers filed in the underlying litigation. See my proposed answer/explanation (the first part of the answer is just about the first meaning -- concluding a written agreement -- but the other issue is covered after that).
AllegroTrans, I agree with your proposed translations (from the reference link) -- written submissions -- but I think the main requirement is that it be clear that we're talking about a written document that was filed in the litigation. I suggested using the word "filed" to make that clear (i.e. "conclusions filed"). "Written submissions filed on X date" works too.
That said, from the description of what "conclusions" contain in FR civil litigation, the closest translation may be "brief" (since it contains a recitation of the facts, a legal argument, and a proposed conclusion that you want the judge to reach):
https://www.litige.fr/definitions/conclusions-avocat-ecritur...