Coolant level and blue temp light

I would say it is possible an overheated engine which is not coolling the transmission can trigger a lot of lights. Many transmissons are heated and coolled by a line which goes to the radiator of the engine cooling system.

Hello, if the water in the radiator was quite low, before topping up, there could be an airlock in the system. Does the internal heater blow hot air? If it only blows cold air, even when the control setting is set to hot this indicates either low water level or air lock in the heater matrix or pipe feeds to it. Quite a few models have bleed valves on the pipe runs under the bonnet just before the bulkhead.
Hope this helps you.

disconect battery wait 5 minutes reconect after 3rd time driveing light will reset itself

not the pump but air in the system or a broken wire to the coolant level sensor
that sensor at the top of the radiator tank has to be completely immersed in coolant or the circuit is broken and the light comes on
same effect id the wire to that sensor is broken in the insulation

possible clean sensor on tank or replace. its probably just one of those things, if coolant level is visually good keep an eye on it as you drive a couple days. also watch temp gage for normal temp.

you put transmission fluid in your radiator? You should have antifreeze in your radiator. You could have a bad thermostat that's causing the car to over heat. Does your temp gauge read a high temperature?

Well, top off the overflow tank to the marks as indicated based on engine temperature, and see if it happens again, and note how long it took for it to happen.

well blue is OK when thermostat opens it should go off this means the car is at operating temp as long you do not have orange or red temp light should be OK hope this helps

your buick century has a low coolant level sensor located in the passenger side of the radiator , sometiomes the sensor get covered with coolant sludge and will show that the cooling system is not filled , sometime the radiator is just still low on coolant . check the radiator when the engine is cold ( IF HOT WAIT !!! 30 mins ) and make sure the level is above the sensor.

If your coolant level is dropping, I would suggest that you take a good look at the water pump. That is a very common coolant leakage probelm area on your vehicle. I change 2-3 a week.
Hope this helps.