A 96 has two fans. Both run at the same time. The fans have two speeds low and high. When low fans are commanded the fans are wired in series. When high fans are commanded they are wired in parallel. There are three fan relays which operate the fans and which are controlled by the ECM.
"Low fans" is commanded by the ECM when either: certain DTCs are set, ECT is above 219°F, engine oil temp is above 270°F or A/C head pressure is above 189 psi. Also, when engine speed is over 3500 rpm and oil temperature is over 261°F low fans will come on. The fans will go off once CT drops about 11°F. If A/C head pressure kicks the fans on, it must drop to 150psi before the ECM will shut them off. Min. on time is about 50 seconds
"High fans" is commanded by the ECM when either: certain DTCs are set, ECT is above 228°F, engine oil temp is above 277°F or A/C head pressure is above 225-psi. Also, when engine speed is over 3500 rpm and oil temp is over 266°F, the ECM will request high fans. The ECM will turn off high fans once coolant temp drops about 11° or A/C head pressure drops below 189 psi.
If you saw 260 on the digital display and the coolant boiled over, obviously the car was way overheating. If you ran the car like that for any length of time, engine damage is possible. Hopefully that's not the case.
I'd be looking for cooling system problems other than just the fans such as restricted air flow through the cooling stack. Also, overheating really stresses cooling system parts such as radiator hoses, heater hoses, belts, radiator caps and so forth. Considering the car is 15 years old, if those parts are original, I'd carefully inspect them for damage.
But, again, the best way to troubleshoot the cooling fans on a 90-96 is using the diagnostic table in the Service Manual.