So from what I've found, the ECU handles voltage sensing and determines if there is a fail state for the charging system. From what I understand, there are low/high thresholds, and from what I was able to find quickly, they are typical 13.6~15v. If that's the case for your car, you should be within range, but it could be slightly different.
From my experience, sometimes a really bad battery can cause a light when it creates too much voltage drop, however this battery usually can't start the car either. Also, with a Ford, I had a belt snap completely (no alternator charge at all) and it took a few minutes of running for the battery light to come on. So I think (at least with the '90s explorer) the sensor was detecting overall circuit voltage, not isolating the alternator.
A few things to check are:
belt properly tensioned and in good condition
alternator mounting bolts fully tightened and secure (no ability to pivot or loosen)
all terminals and connectors to/from alternator fully seated and in good condition
terminals to/from the battery are secure and have clean contact surfaces without corrosion or rust
connect multimeter leads at battery and position the multimeter so you can read it from the driver's seat. Then see what the voltage reads when you hold the motor at 2000-3500 rpm. Try to sample a few different speeds, but also try to hold each sample for a few seconds. (BE CAREFUL! Watch where you stick your hands and where you lay the leads! Lots of hot or moving parts in there so try to drape them outside the engine compartment if you can and keep your hands clear of the engine and fans when it's running. I don't mean this warning to sound demeaning if you know what you're doing, just fair caution. )
There could be something totally different causing the issue (like a damaged or shorted component) but these are some simple things to check.
Also, it is totally possible that the belt is loose or in bad shape, and thus losses friction at higher rpm. This would explain why it only happens at mid range rpm. At low rpm it holds friction and generates sufficient voltage. At high rpm, it does give out a bit, but the surplus of voltage generated can be regulated down to an acceptable level by the VRR. At mid range is just the right range to keep functioning, for a while, but not enough to be happy. While running high power accessories, the alternator has to supply more voltage to keep the circuit charged because of the draw of those accessories, ergo why it is more common with these accessories on. Also, if 13.7v was your idle reading, then it could even be slipping a bit there to.
Finally, I don't understand the way in which the ECU detects or reports voltage, so I'd stick to using a multimeter that you know is in good working shape for your testing.
