Question

Tosafot at the beginning of tractate Bava Metzia, on the words “and this one takes a quarter,” brings the view that in a case where a person stated that his document is invalid, and had he not said so the document would have been accepted, there is no migo here based on the fact that he could have remained silent. The reason is that this migo would be used to uphold the original claim as true, but here the original claim is false, since in reality the document is forged, and you are trying to uphold a forged document through a migo that you could have remained silent—and that is not a migo.
And it is difficult for me: why is this called using a migo to uphold a false claim? After all, my main claim is not that the document is genuine, but that this document is a way to prove that I am believed. If so, what I am trying to uphold through the migo is not that the document should work in my favor, but rather my credibility through the migo. So it turns out that what I am upholding through the migo is something true—namely, my credibility. So why does Tosafot write that what the migo comes to uphold is the document, and therefore write that it is upholding something incorrect?