It took me a while to understand why 208 was 208 and 240 was 240; Basically in a residential system you're splitting 1 phase 180 degrees. So the peak voltage difference between L1 and L2 will be 240. Most commercial buildings receive 3 phase power. For '240v' applications they use L1 & L2 or L2 & L3 or L3 & L1. The voltage difference between any phase and ground will be ~120v just like split-phase residential. BUT; the phase difference between L1 and L2 isn't 180 degrees... it's 120. So the voltage difference will be ~208v not ~240. The connector can only give you what's available...
Actually, that isn't quite accurate but it's close.
The power company could give you lots of different voltages, it's just down to the service that they can deliver, sell and you can use. Technically, in a three-phase installation, they could offer you 120V *and* 240V, it's just that it comes with some restrictions or power balancing issues.
For example, some older commercial installations offer split-phase delta service on three phase. 120V is provided by grounding a transformer center tap from L1-L3 on a 240V delta configuation (where L1-L2, L2-L3, and L3-L1 are all 240V). This works for smaller installations, but runs into some issues in that 120V loads are concentrated on only one of the phases and you can have significant imbalance on L1-L2 and L2-L3. This is also the source of a lot of handyman boo-boos as the voltage between ground/neutral and L2 is 208V (the "high leg" or "stinger").
The reason that 208Y120V is pretty much standard and popular is that the neutral/ground is located in the "center" and 120V loads can be balanced across any of the phases; switching power supplies have made it easier to attach electronics to nearly any voltage; and resistive loads are easy to adjust for 208 operation.
I'd probably argue the semantics of saying that split-phase is splitting 1 phase 180 degrees, because it really doesn't reflect what's happening if you consider power in the context of phase windings in a turbine generator. But you're close enough that it's not completely incorrect, either.
