It really depends on the ISP and what their upstream setup is. You will need the generator to power your cable modem and router, although some ISPs include backup batteries in their cable modems (esp. if the modem is designed to also provide a phone line) if you bring your own modem it won’t have one, and your router will need power too.
Assuming your equipment stays powered on whether or not your internet will continue working is not very predictable. Recently a car hit a transformer and took out power a few blocks from our house but not our power, however my ISPs equipment was in the segment that lost power and we lost internet until that segment was repaired. It’s likely my ISP had no backup power for their field equipment when it lost power. This isn’t very surprising since most users won’t have backup power at home and won’t be able to use their home internet when power is out anyway.
OSI Layers are bullshit leftovers from the past, they’re no longer applicable in modern computing.