I don't think that has a name, although "moving the goal posts" would be close because they started out saying that something IS proven, which implies anyone could repeat the experiment and come to the same conclusion, then change to "let them believe", a much weaker claim and something you didn't argue against. (I should add that we should argue against letting believe crap like that because irrationality in any form hurts everyone).
When I was a big time EST
guy, we often talked about prostelytizing, although we called it "enrolling". We noted this behavior in people we were trying to convince. We would begin by "sharing" our experience and at first they would be curious, because we were just telling a story. If they became interested in participating themselves, they would say, "don't prostelytize". Whether or not we were prostelytizing is not the point.
In the case of this Noah believer, you are taking the stance that "proof" requires some evidence. However you present that in your discussion, forcefully, or simply stating that you are a sceptic and require some data, at some point the believer realizes they aren't getting anywhere. Depending on where they are at, they are probably starting to think that this whole "evidence" thing might have some merit. They are getting a little spark of doubt. Defense mechanisms immediately pop up and they have to blame you for not "allowing" them to hold their irrational belief. You may need to look in psychology books for a definition of this instead of lists of logical fallacies.