Mostly correct. It’s not technically in the Geneva Conventions, but The Hague Convention. They all sort of get lumped together in common parlance. The Hague convention doesn’t dictate FMJ. It prohibits explosive or expanding bullets (among a host of other things). The “target” for the original conventions were the British soft points made at the Dumdum factory in India.
At least during my deployments, we were careful to distinguish between the NATO ISAF counterinsurgency mission and the OEF Counter Terrorism mission. Troops supporting the ISAF mission were issued with “Geneva-compliant” 77-grain OTM ammunition, because COIN is a military mission. The slight hollow tip on the OTM wasn’t supposed to be designed to expand, but was purely designed for superior ballistics.
Troops supporting the OEF CT mission were not engaged in warfare, so they got to use the more effective 77-grain TMK ammunition. As a battalion, regimental, and SPMAGTF SJA, I was told to keep an eye out for and advise against my Marines getting their hands on CT ammunition.