cold engine with high temperature code

Do you have an OBD2 scanner that you can use to check for trouble codes?

Based on your symptoms, there can be a short in the engine coolant temperature sensor circuit OR the engine coolant temperature sensor could be bad.

EDIT:
There is a second temperature sensor for the exhaust heat recovery system that can also turn on the high coolant temperature warning light. So, there could be a short in that circuit OR that temperature sensor could be bad.

I would first look for a short in both circuits because the temperature sensors very rarely go bad.

 

Using a scanner to read the trouble code should determine which temperature sensor/circuit is causing the problem.

 

If you have a friend with a similar aged Prius you can swap out parts in the temp sensor circuit until you find the part that has failed.

 

Parts cannon is not needed.

A good scanner can tell you if either temperature sensor (both are very reliable ntc thermistors) or their wiring is

a) open (cut or disconnected)

b) shorted (to its own pair of wires, other wires or ground)

and

c) can read out the actual temperatures which should be close to ambient when the engine is cold.

Live Temperature Readings