You are almost right, but wrong.

The cooling fan runs:
... At a minimal speed anytime the A/C is turn on, this is to draw some air across the condenser to cool it.
... At a minimal speed when engine temperature reaches something around 200F, should the temperature continue to increase, the fan will speed up in an effort to control coolant temperature.
... NEVER with engine temperature below 200F and the A/C turned off.


The design is simple:
The ECM and the FCM carry on a conversation ANYTIME the key is 'on'. Every 200 or 400 milliseconds (I once knew exactly, but forgot) the ECM says "Are you there?" The FCM responds with "I'm here".
If the ECM sees engine temperature rising above around 200F, it tells the FCM to run the fan at slow speed (it does this by modifying the "Are you there?" message with a longer pulse which indicates the fan should run); if the A/C is turned on, it tells the FCM to run the fan at slow speed. If the coolant temperature rises, the ECM tells the FCM to speed up the fan by making it's pulses to the FCM even longer.

If the ECM does not get an "I'm ok" response, it will turn on the "low coolant light".
If the FCM does not hear from the ECM, it speeds the fan up to Warp Speed (to protect the engine because it does not know the temperature of said engine).

If you have not "Low coolant" light, then the ECM thinks all is OK. This means the FCM is responding and the ECM sees the response.
SO, either the ECM is not telling the FCM to run the fan, or the FCM is just not doing it's job.

Check the 50amp fuse to the FCM (in small fuse box next to battery).
if it's ok, then:
Replace FCM.

I have a hard time beleiveing the ECM is the cause of this but it could be. But tha'ts around a $2000 repair IF you can find a new ECM. If it goes that far, yank the ECM and send it to one of the companies that work on them.