So in this case, if the pitcher had said something like "The batter missed 1B" instead of "he missed 1B" that would have been OK, I'm guessing. What if the pitcher had said it just the way he did (using only "he" as a reference to the batter) but he pointed at the player in question - would that physical gesture suffice to satisfy the rule of properly specifying which runner you are appealing?
I agree with you that this seemed like a "nit not worth picking". I actually got the impression that the ump wasn't paying attention to the batter when the batter rounded first - there were runners on 2nd and 3rd when the play started, and they both came around to score, and there was a throw to the plate to try and get the guy scoring from 2nd (though it wasn't a close play at the plate, at all). Point is, there were other things to be keeping an eye on as the play unfolded, and the ump was working this game by himself. I don't think he saw what happened at 1B, and I think he chose to "pick this nit" instead of just admitting that he didn't see what happened.
Ultimately it didn't have an impact on the final outcome, but it was the first time I'd seen this call made, and I wanted to clarify it to make sure my own team appeals correctly in the future. Sounds like the bottom line is: be specific about the player you're referencing, and be specific about which of his (or her) actions you're appealing.