Depends on the study design. Infections are detected by doing a PCR covid test. However, different studies might have different criteria for exactly who gets tested.

PCR doesn’t differentiate between live and dead virus molecules, which makes it less than diagnostic for active infection without symptoms, right?

That distinction isn't as important for evaluating vaccine efficacy. It wants to measure if the person was infected, not if the person is infectious.

If a trial subject is exposed to virus and the vaccine works to trigger an effective immune response, how does PCR know the difference between virus molecules neutralized by vaccine response and those neutralized otherwise?

In those cases the virus particles don't replicate enough to be detected.

How is a difference be determined between between vaccine enhanced immune response, normal immune response and not exposed?

The basic structure of a phase 3 vaccine efficacy trial is that you give the vaccine to half of the volunteers and placebo to the other half. Then you you wait for them to naturally be exposed to COVID. When a pre-determined number of infections is reached, the data is unblinded and they compare how many infections happened in the vaccine group vs the placebo group. If less people are infected in the vaccinated group then it shows that the vaccine is effective.

The initial comment wrote:

The FDA initially wanted 2 months of data for under 12, just like adults. Then they changed their minds and wanted 6 months instead. I'm sure they had a reason, but in 70 years of vaccines, we've never had a side effect after 2 months. If they changed their minds once, they can damn well do it again and get this thing out there.

If child mortality and clinically significant infection are low, then would it not potentially take longer to see a statistically difference in a difference between vaccine and placebo subjects?