I agree, there is no "official" definition that revolves around the method of connection.
But this might help. There really isn't a "one piece" starter in NEMA designs either; the overload is always a separate device. There are just 2 ways of connecting it to the contactor; close coupled, i.e. with stabs that connect directly to the contactor terminals; or stand alone, where the user must make connections with wire between them. To add to the mix, when you get above NEMA size 4, the overload is completely separate because the power must go through CTs, and only the CT secondaries go through the OLR.
All of these are "starters", because if it is used on a motor circuit, it is a starter. If it is used on something else like a heater element, you don't need the OLR, so it is just a contactor.
