No, anything on the belt that seizes will slow the belt down, so the alternator seizing will slow down the W/P too. The fact that all the problems disappear with the headlights off pretty much means if the belt is good and the tensioner has tension, the alternator has to be the culprit. Unless of course the OP is giving us an incomplete description, but I'm assuming he isn't. The problem in the alternator could certainly be heat-related, and that's something the test machines in the parts stores don't check for. They just load it and run it up to speed to check for output. Running with a full electrical load generates plenty of internal heat in any alternator, but the testers don't run it long enough for problems like that to show up.
Vstar,
That sound's pretty good to me, as the Water Pump, Alternator, and Crank are on that belt according to the Service Manual for a '93 KA24E 4cyl. I would also like to know about the Water Pump just to rule it out completely, as in Leaking or Binding, and as an Elementary makeing sure that the Belt has the Proper Tension, and not Glazed, causeing it to Slip and Squeal under extra load of the Alternator.
I would also like to know the Battery condition, Age, and if the Terminal's are Clean and Tight to illiminate excessive draw on the Alternator. This is a common area that is Neglected till a problem pops. I don't know the Region of this Truck and the Winter Temperature's, but many area's in the US are in Severe Cold Temps which put's a lot of stress on the Battery and the need for Clean/Tight Connection's.
We might be surprised if it was the Battery, Ground Strap, corroded terminal's, the Belt itself or not properly tensioned.
Sometime's Simple isn't so Simple, but We are Glad when it is.
Regards,