There's no sense in quibbling over a vague and imprecise summary of the concept. Legally, fraud requires an injured party. In our scenario here, nobody was injured, so no fraud. Whether temporarily possessing a document to hand it to its intended recipient counts as "obtaining" is not really the pertinent question.

I suspect it may fall afoul of a law about making false statements to the government, but that's distinct from fraud.

That's obviously nonsense. Lying on a loan application is fraud, even if you don't get the loan. You're confusing criminal liability with civil liability, which does require damages.