C13 engine coolant temp and fan issue

I have a 2005 pete 378 with a c13. I’m having a issue where the gauge in the cab says the truck is running 200-225. I’m scared to let it keep running hotter just to see if the fan is coming on. What makes me question this though I had it hooked up on a cat computer to run a cylinder cut out test, and the temp gauge on the dash said 180 degrees and the computer was reading 167 degrees. Which in that case I drove the truck about 10-15 minutes to the shop. If the truck was actually trying to run hot wouldn’t the fan come on at cats desired 217? Also the cab gauge obviously was reading different than the ecm by 13 degrees so wouldn’t the temp gauge have to show 230 degrees for the fan to really come on at 217? How do I fix this problem? the fan in this truck has never cycled right in my opinion unless the ac is on.

 

Ten four will do I have one I keep in my truck

 

I was thinking of putting a mechanical gauge in it

 

Does the temp sensor hooked to the ecm actually control the guage