In my district, we had two breaks in elementary school: a fifteen-minute recess and a 45-minute lunch break. The students go outside and play during both breaks. Teachers look after the kids during recess; at lunch, that's done by a rotation of parent supervisors.
In high school, there were four classes a day, in different classrooms. Students got a five-minute break to move between their first and second classes. Between the second and third classes, there was a fifteen-minute long break. Lunch break was an hour between third and fourth classes. In all cases, students didn't have to go outside, so a rotation of teachers roamed the halls making sure nothing bad happened.
My teachers usually referred to this as supervision or being on duty, as in
"I can't meet then. I have supervision tomorrow."
"I'm on duty (at recess / after third period / etc.)"
"Who else is on duty today?"
I come from a family of teachers, so I can confirm that these are still common usage in my district. However, I'd imagine that usage differs from place to place, especially if a specific phrase like lunchtime supervision or recess duty is used in your collective bargaining agreement. Any combination of
playground / lunch / recess / break / ...
supervision / monitoring / duty
will give you an understandable term for this situation which I guarantee is widely used somewhere; the examples playground duty, playground monitoring, and lunch monitoring, are all cited in other answers. (Note that none of these take the article a; you'd say "I have playground duty" instead of "I have a playground duty.") So
"I have break duty after my 4th lesson on second floor"
sounds good to me!
Your best answer for what you should call it, though comes from Mitch's comment: use whatever phrase is most commonly used by the teachers around you!