That makes sense, but it still does not differentiate between the "may" used in the green arrow situation and the "may" used in the right turn on red section. In both of those sections, the driver MAY proceed, after yielding to pedestrians and other traffic that has the right to be in the intersection. The only real difference is that, with a right turn on red, the vehicle has to stop first.
Regarding a right turn on red: "vehicular traffic traveling in the travel lane nearest the right hand curb or other defined edge of the roadway, unless a sign has been erected in the appropriate place prohibiting this movement, may cautiously enter the intersection to make a right turn onto a two-way street or onto another one-way street on which all the traffic is moving to such vehicle's right after such vehicle has stopped as required in this subdivision and yielded the right-of-way to pedestrians lawfully within an adjacent crosswalk and to other traffic lawfully using the intersection."
Regarding a green arrow: "Vehicular traffic facing a green arrow signal, shown alone or in combination with another indication, may cautiously enter the intersection only to make the movement indicated by such arrow, or such other movement as is permitted by other indications shown at the same time, but such vehicular traffic shall yield the right-of-way to pedestrians lawfully within a crosswalk and to other traffic lawfully within the intersection."
Let's say you're at an intersection where you can't turn right on red, but there is a green arrow to turn right that either comes on before the full green, or stays on longer than the full green. I doubt many people would agree that's it's okay to not proceed on the green arrow, but to choose instead to wait for the full green light. Yet the language used in the statute is just as permissive vs. obligatory in both scenarios.
(By the way, I understand that the OP did not proceed because she did not think it safe. Just playing devil's advocate here for the people who have stated that it's okay not to turn right on red even when traffic is clear because there's no requirement that you do so.