"No reentry" is not the same as "emergency exit only"

A week or so back I discovered that one of the school's exit doors was locked or jammed shut, and alerted the security department. I was in the same building again today, and decided to see if this had been addressed.

It had been. I had to give the door a shove, but it opened into a stairwell. The door was labeled "No reentry on any floor," and had no knob or handle on the stairwell side. But the stair looked to lead out to the parking lot, though, which was where I wanted to go, so through the door and down the stair I went.

Two flights down I found the door to the outside, equipped with a big latch that read "EMERGENCY EXIT ONLY: PUSH HERE TO EXIT. ALARM WILL SOUND." After a brief debate as to whether I should just open the door anyway, I retreated back up the stairs, pulling at the knobless doors until I found a level with someone still working, and I pounded on the door until they let me in.

And this is the thing that really gets me: on witnessing my release, a woman came up to me and proceeded to lecture me on how "no reentry" means "emergency exit only."

I wonder how this woman ever gets out of the subway system, where half of the exits are no reentry and the other half require payment to get back in.