"The overhead bins are for large carry-ons only" does not make sense. Suppose I bring two personal-size items. Each of them fits under the seat, but they do not fit under the seat together. This is definitely in-policy, I can declare one of them to be a carry-on, as the carry-on rules restrict only the maximum bad size.
The policy should be that overhead bins are for carry-ons only, and pax decide which of the qualifying items is a carry-on. As a cutesy to others, I may decide to put my only carry-on item under the seat, but this is for me to decide.