I read that when the ECU tries to start the engine, it cranks the engine for a second without injecting fuel to get the RPMs up high enough to prevent a rough start. It almost seems like the RPM cannot get high enough in cold weather before that 1 second passes, causing the engine to aggressively shake for a second. When the RPM is too low, you will get major engine misfire which causes the engine to violently shake. When my Camry did this, I could tell with some confidence that the RPM was momentarily much too low to correctly start.

I am 95% sure that this does not represent a mechanical failure (or pending failure) in the engine.

I had this shake happen to me in my 1984 Corolla when the choke was stuck off in the cold. The engine would start to stall, I'd hit the gas to recover, and the engine would shake violently for a second as it recovered.