Overheating heats rapidly upon starting,indicator light reads low coolant,but levels are where they are suppose to be?

It sounds like you may be experiencing an issue with the temperature gauge in your Buick Lucerne, as well as possibly an issue with the engine. If the temperature gauge is reading below the "C" (which likely stands for "Cold"), it could indicate that the engine is not reaching normal operating temperature, which could be caused by a number of factors such as a low coolant level, a faulty thermostat, or a malfunctioning water pump.
If the engine light has also come on, this could indicate that there is a problem with the vehicle's emission control system or another component of the vehicle. It is important to address this issue as soon as possible, as driving the vehicle with an overheated engine or other malfunction could potentially cause further damage.
To troubleshoot the problem, you should first check the coolant level in the vehicle to make sure it is at the correct level. If the coolant level is low, you should add more coolant to the system according to the instructions in the owner's manual. If the coolant level is normal and the temperature gauge is still reading low, you should have the vehicle inspected by a qualified mechanic as soon as possible to diagnose and repair the problem.

Depends on how the system is supposed to work. I am not familiar with them.
If there is a device to warn of low coolant level it might be a resistance-based device or a small float with embedded magnet and a reed switch which is a common way of achieving the result - sometimes the float will stick...

If there is a coolant temperature switch, these are usually very reliable and like a switch it is either on or off. Sometimes these are fitted in pairs to indicate if the engine is running too cool or too hot.

A switch is unlikely if the engine has computer control - in this case the unit, the temperature transmitter, sender or sensor is a thermistor or a resistor that has a value that varies with temperature. It will need to have a resistance that falls in a specific range at a specific temperature. In the case of an overheating engine the computer will switch on the light.

I suggest you read the operator's handbook to determine the precise function of the coolant light.

Sensor in the fill tank is probably bad .
LOW COOLANT
The IPC illuminates the LOW COOLANT indicator in the message center when the IPC detects a low coolant level condition from the PCM. The IPC receives a class 2 message from the PCM indicating the coolant level.
Coolant Level Control (If Equipped)
The engine cooling system contains an engine coolant level switch to alert the driver in the event of a coolant loss. The powertrain control module (PCM) sends out a coolant loss signal over the coolant level switch signal circuit. When the engine coolant level switch reads a low coolant level in the fill tank, the switch opens. The message center receives its power from engine wiring harness junction block on the battery positive voltage circuit. Ground is provided by the ground circuits via the body wiring harness junction block and the engine wiring harness junction block. The cluster receives the class 2 message from the PCM indicating Low Coolant and displays the LOW COOLANT LEVEL message on the driver information center (DIC).

Is the fill tank full ?

If by the red light comes on upon cold startup, and the coolant level is fine, and it hasn't had time to get hot for said red light to indicate hot, then it's a stuck sending unit. If it always reads hot, the computer won't put it in cold start mode because it thinks the engine is already warmed up. That's the way it works. And especially if it isn't pushing any coolant from the radiator and over filling the reservoir, then it isn't over heating. IF it is overheating, I would first check that the fans are working properly and if they are, then I would go to the thermostat. And THAT means getting your hands dirty. You have to drain the coolant down just below the thermostat level, unbolt the thermostat housing, replace the thermostat and gasket and put it back together. The re-fill the radiator and reservoir with what you drained out.

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.

There may be a thermostat problem, but your symtoms indicate low coolant level. When coolant is too low, it will not circulate, so you get no heat in car, and the coolant remaining in engine block overheats because no cool fluid coming in.
You need to fill the system up properly then look for leaks.
With cold engine, open radiator cap. Fill with antifreeze directly in radiator. Start engine, top up rad if level drops. Put heat control on Hot. Fan on low, heat on high. Gently rev engine and top up rad as needed. IF as engine heats up, it starts kicking fuild violently out of radiator, then the thermostat is bad. if not, then recap radiator, top up overflow bottle and you're done. Look under car for any sign of leaks. Steady dripping indicates a leak, then address that.

I read in the manual that sometimes you need to add or check the radiator as well as the coolant level, I had filled just the reservoir and that did not fix the low coolant light. However I did read that when filling the radiator there is a screw on top of the thermostat housing to bleed the coolant.... you are supposed to start the engine to get the air out of the line as you fill the radiator. Of course only do this when engine is cold. I still have to do this but I cannot get the radiator cap off as I have arthritis and it is very difficult. I hope this helps. Check the manual also.

CHECK GAGES light (1) indicates a potential engine failure(low oil pressure, low coolant, coolant overheating) has occurred.
· Check gages and shut down engine. Gages should read as follows: Oil pressure 5 to 10 psi (34 to 69 kPa) at idle; Water Temp gage below 230 degrees (110 C).
· Check engine oil level (Para 2-11) if OIL PRESS gage indicates low reading. Check coolant level (Para 2-11) if Water Temp gage indicates high reading. Notify Unit Maintenance if levels are low.

If CHECK ENGINE light (2) is on, check engine oil pressure gage (3) and water temperature gage (4). If both indicate normal, proceed with mission and then notify Unit Maintenance. If either one does not indicate normal, stop engine immediately and notify Unit Maintenance

If the temp gauge is reading normal, the lights are flashing, and the coolant level is full, the level sensor in the coolant bottle is the problem. I'm a service consultant for a Saturn dealer and I've run across this many times.

If the engine is overheating, that's why the light is on. Its a P0271 code I beleive. Its a "OVERHEATING" code.Dont drive the car. You are loosing coolant or its out of coolant, check the coolant level in the overflow bottle. We have seen some headgasket issues on that model car. I would remove the spark plugs and look into the cylinder for any signs of coolant on top of the pistons. Let me know what you find.