Or, for instance in the case of particle physics, it means the probability you are just looking at background.
Well, technically, the probability that you will end up with a result given that you are just looking at background. I.e. the probability that after the experiment you will end up looking at background thinking it is not background*, assuming it is all background.
if it is used for a threshold for such thinking
It’s really awkward to describe that in English, though, and I just assume that this is what you mean (while Bayesianists assume that you are conflating the two).
Well, technically, the probability that you will end up with a result given that you are just looking at background. I.e. the probability that after the experiment you will end up looking at background thinking it is not background*, assuming it is all background.
if it is used for a threshold for such thinking
It’s really awkward to describe that in English, though, and I just assume that this is what you mean (while Bayesianists assume that you are conflating the two).