i'll tell you a bit about the system since i've worked with them a bit
your ASR system on your 96+ car is controlled entirely by the body control module, not the ecm. the ecm does play a part, but only by getting timing retard signals from the BCM.
if the ECM had failed somehow or the traction control retard wire to the ecm was broken, traction control would still work, although stage one would be considered a fail and it would immediately go into stage two and three (see below). i have unhooked the timing retard wire. ASR still works fine, it's just more aggressive.
it has a four channel wheel speed reluctor system by default, for 96, i think... a reluctor on each wheel. crap on the reluctors or a failing sensor can cause these issues. they are magnetic sensors, metal will stick to them and probably screw stuff up.
it's worth trying to clean them before going any further. they are the same reluctors used for ABS. the ABS and ASR system on your car are fairly integrated.
when rear wheel speed is higher than front (according to the reluctors), it assumes you're doing a burnout.
ASR on the 96-97 cars is a three stage system, different from the ones that just apply brakes.
stage one, timing is retarded. the body control module sends a traction control timing retard signal to the ECM. this is the same signal as the old TCS system used. it retards a certain amount of timing, based on a table in the ECM relating to vehicle speed and rpm i think. at this point the LOW TRAC light doesn't go on, as it's considered just a mild correction.
stage two applies the rear brakes using a solenoid on the ABS module. it only produces a little bit of braking force. the LOW TRAC light should flicker at this point or just stay on, i forget which..
stage three uses the motor and throttle cable thing on top of your drivers side valve cover to force the throttle closed. the LOW TRAC light goes on at this point, and remains on for a few seconds even after traction is restored.
if any problem is detected with ABS or the speed reluctors, the ABS and ASR systems are both disabled for safety, but error checking is primitive and never ends up working. pulling your ABS fuses (ABS IGN and ABS BAT under your hood) disables ASR and ABS at the same time.
if you're having issues with it, do that until they're corrected, as it could easily cause you to get rear ended or lose control in a maneuver. you might end up wanting to just leave them unplugged as both of those systems are for pussies and soccer moms.