I would also add that the inscription is part of the forgery.
Ordinarily, I would agree with you, but in this case, adding the inscription is a material alteration because this now “places” (attempts to place) Moore at the restaurant.
Black’s law dictionary says this is a textbook forgery.
It’s worse than when you realise she must have referenced her divorce papers — long after the supposed initial inscription. That alone is a good argument for forgery as the only reason to add the stamped style signature means she intended to deceive.