Answer

First of all, one can understand the superiority of a definite claim as indeed being due to credibility. It is not a proof, but he is believed with respect to the claim itself, like a miggo as a strengthening of a claim, for example a miggo based on brazenness, which is not a proof but does confer credibility. One could also say that there is a presumption that a person does not sue unless he has a valid claim against the other party, and then there is actually real evidence here. And that is probably what Rashi means in the wording you cited.
As for an uncertain claim, I seem to remember that in Kovetz Shiurim he discusses whether this is a weak claim, and perhaps even actual evidence against the claimant, or whether it is not a claim at all. According to the view that in a case of definite versus uncertain we do not say this in order to extract property from someone in possession — this is a dispute among the Amoraim, and that is also the Jewish law — it seems that it is considered a claim, because otherwise the party in possession could not prevail without any claim at all.