The factory security system is a rather simple device. It's armed and disarmed through a switch attached to the back of the lock cylinders. Not surprising that most don't function properly these days after years of wear and tear. The system has sensors at both doors, hood and hatch and when triggered disables the starter, flashes the headlights and honks an auxiliary horn (it's the one by the brake booster).
In '91 and '92 Toyota offered a supplimental system as an option which added a glass breakage sensor and keyless entry.
With both types to re-enable the starter you must rearm the alarm and then properly disarm it. From what I remember, pulling the battery cable doesn't reset it unless the latching relay inside the security system's control box is no longer latching when thrown... I know it doesn't reset mine, although it's been a while since I've dissected one so I could be remembering things incorrectly.
EDIT: The upgrade kit for the '91 and '92 isn't plug and play for older models, nor has it been available for years.
The stock alarm ONLY uses switches, these monitor locks, doors, hatch and hood. If someone breaks the glass but doesn't unlock the door or open the trunk, it does nothing.
I too highly recommend investing in an aftermarket alarm. Me, I've kept both and they're thoroughly integrated with eachother. I also recommend getting an alarm with paging capabilities.... or one that screams in a female voice when triggered.
