May be no cooling system problem at all, except of course, faulty sending and/or gauge as suggested above. Do you have access to an infrared temperature sensor with LED pointer? If so allow the temp to rise, stop the car, and place the LED pointer on the radiator hose or block, preferably away from the exhaust manifold. I own one of those guns and find it is the easiest way to check temps. If you find the temp to measure 200 degrees F +/- 15 degrees F, or so, then you know you have a "sensing" or "reading" problem. If so, I'd start with the temp sending unit first, as noted above.
2001 SSEi- black; stock 92K
1995 Vette; Competition Yellow; stock; 38K