No they are not "private property". So in that sense, she had her vocabulary wrong. But while it may be "public property", yet the govt. can set up rules of usage. For example: A library may be "public property", yet they can set opening and closing hours. If you go in through a window at 2am at a closed library, you can be guilty of "trespassing" (yes, on public property). And all sorts of other "rules" for the public good, on public property (ie.: no dogs allowed, no alcohol, no motorized vehicles on the grass, etc...).
So here's what would happen if you challenged her usage of words: While you might successfully show her that it is public property, she could just turn around try to tell you that you're improperly using it (for whatever lame-*ss reason she wants to try to morph to apply to you). And then you just enter into a p*ssing match of whether or not you have really violated any usage rules. It may be entirely true that the school yards are not open to public usage. ie.: students only or whatever. Starting about 25 or 30 yrs. ago, fences started popping up at public schools all over the united states, with signs (that no one pays attention to BTW) stating something like "permission to pass revokable" or "check in at school office" or something like that. But there is usually still an always-propped open turn-style gate, and people still jog the track, play basketball, or whatever. In other words, the sign is there so they can shoo away perverts, or to keep from getting sued if you fall off the swingset and try to sue them (they can say "well you shouldn't have been there", etc...).
Most of the time you can still detect these schools till your heart's content, just as any other type after-school user seems to still do so. But as you know, detecting is an odd-ball hobby, that someone might think is damaging (perhaps she saw you in the middle of a retreival and didn't know you won't leave any traces). You can try to debate her on that point, but you know where that will probably go.
The best bet is to be more discreet, and go at more off-hours, and just avoid her shift in the future.