Car/truck batteries last 5 to 6 years, then they run out of chemicals to hold voltage, the internal plates also deteriorate
You test battery voltage after its been sitting for 4 hours at least, longer is better
New battery will have 12.8volts to 13.0volts
3 year old battery will have 12.5v
5/6 year old battery will have 12.3v, and time to shop for battery sales
12.2v or less is a done battery, replace now
As the outside temperature drops below 45degF the batteries AMPs/Volts also drop with it
So not at all unusual for an older battery to work fine when warm then not work when cold
Test your battery voltage on the next cold morning
Just FYI, the computer has nothing to do with starter motor working in a 1996 Ranger
In 1999 V6 Rangers got PATS, which can prevent starter motor from working, but you would see a flashing THEFT light on dash