If tomorrow night I go outside and see lots of bright lights in the sky, and a crop circle the next morning (which is especially weird because there didn't used to be any crops there at all), and the news reports that lots of other people have seen the same thing and the government is passing it off as a sighting of Venus, then that's somewhat more likely in UAP-world than ¬UAP-world.

This example suggests that you're confusing P(OU1|UAP) with P(UAP|OU1). To determine P(OU1|UAP), image you live in a world where UAP is true.

Unfortunately, the analysis so far hasn't been clear on what we mean by P(UAP): does it mean the probability that there are alien visitors within Earth's atmosphere or that their ship is flying over head right now?

Assuming the former, to estimate P(OU1|UAP) assume there are aliens on Earth, if that is the case what's the probability of you observing the light in the sky. Obviously this is hard to estimate but one would start by speculating about the potential motives and behavior of the aliens.