I think nmno's response pretty much sums it up. Except, I don't think there really is a "straight answer" - each company and each show is different. The way props are integrated with the set is true. But when you get down to the fine points (where does it go on the pre-show checklist? which IATSE crew handles the item?), what seemed clear can become confusing.
I have run into this question on plenty of shows at all levels of theatre production (academic and community right up through large professional houses). Sometimes the lines are clear enough for most of the show, until you finish load-in and realize that the props master thought the set designer was handling, say, the wall dressing (pictures, curtains, etc) and the set designer thought the props master was on top of that. The other side of the spectrum has happened when there is a huge prop - like, say, a car or big wagon - that needs to be engineered and built along with the set. But is it a set piece or a prop? Ultimately, to me and the teams I have worked with, it doesn't matter how you categorize it, as long as someone is identified as the person responsible for each item. A good production manager will see these issues miles away and communicate it all to you so these questions are under control well in advance. Like with so many parts of our job, the trick is thinking ahead and communicating clearly across the whole production team.
