Can undergraduate admissions officers tell if you aren't actually a good coder?

No, short answer.

This isn't an issue with coding, it's simply one of more modern things that people have started lying about on their CVs. Of course we still have the good old summer placement descriptions where "typing numbers into Excel" magically becomes "solving serious engineering challenges".

Ultimately if you're reviewing this sort of thing, you learn to use judgement about who is telling the truth and who isn't. Most of the time you can't and you go by your gut instinct.

At a recent summer camp I help out at, we had maybe 20 students doing progrmaming projects. Around 15 claimed some experience with programming either in their application or through email correspondence before the camp. I think perhaps two managed to write a FizzBuzz program in under ten minutes?

I doubt this is something MIT is immune from. It is a problem which particularly plagues undergraduate admissions where students are pressured into making the most boring events sound spectactular in order to compete with about a thousand other people. This is why almost all the top institutions interview.

I once spoke to an admissions tutor at Oxford (Physics) and he said that essentially they ignore the application. They'll interview you if your grades are good enough and go solely on that. If your grades are close, but your application is great then they'll consider it. They introduced entrance exams, but even they were a poor performance indicator because rich kids pay tutors to help them prepare which skews the results.

Tom Stagliano's makes the point. It doesn't matter. Coding is a tool to use at the university level, not the goal of education. In addition, coding can be learned at the university and is not necessary for admission.