how do I count that last number? Is it the second beat of the bar, or literally the second note printed in the bar? The former felt more logical to me except that I then couldn't work out how you would notate something that happened between two beats.
I agree that the second beat is the more likely interpretation. Another argument in favor of this interpretation is that while the "beat" interpretation might not allow a cue between beats (but see Michael Seifert's comment), the "note" interpretation would make it impossible (or at least cumbersome) to place cues between notes, for example if the entire ensemble is holding a single whole note for the duration of an entire measure. The interpretation that the
2 refers to the second note in the bar places an unnecessary restriction on the designer, requiring cues to be scheduled in line with the rhythm of the music.
On top of that, what if there's a duet or other ensemble? Which part's second note is intended?
For all that, though, I wouldn't rely on this analysis or on any other assumption. Just ask. From a comment:
If we give you the wrong answer here and you act based on wrong information, it will make you look a lot worse than admitting you don't already know this particular numbering system.
You can ask the question in different ways, of course, some of which might put you in a better light than others. The Workplace is a better place to ask about those, but I'll suggest something like "I've never seen this system before, so I want to confirm my understanding that the fourth number, if present, refers to the beat. I see that this makes sense for cue 328, but I wanted to be sure" or, if applicable, "I don't know what to make of the fourth number, because if I assume that it refers to the beat then cue 328 doesn't seem to make sense."
The advantage of this approach is that it shows you've applied your intelligence to the problem but are humble enough not to assume that you're right and cautious enough to place the well-being of the project above your own image and reputation. Such an approach would not diminish your stature, in my eyes at least, but would only increase it.