Just so we are on the same page, your 2001 has the 2.5L SOHC Lima 4cyl engine.
In mid-2001 Ford switched Rangers over to the 2.3l DOHC Mazda L engine, Ford called it a Duratec engine, but Ford called quite a few engines Duratec, lol, I guess they liked that name.
So that can be confusing for 2001 model year
All 2000 Rangers would have the 2.5l SOHC Lima
All 2002 Rangers would have the 2.3l DOHC Duratec
All fuel injected engines had 2 coolant temp reading devices, until mid 2000's.
ECT(engine coolant temp) sensor, 2 wires, only used by the computer
ECT(engine coolant temp) sender, 1 wire, only used by dash board gauge
On the 2.5l the sender is on drivers side toward the back of the engine, near oil pressure sender.
Drawing here: | Repair Guides | Sending Units | Coolant Temperature Sender | AutoZone.com
Make sure you got the right part, but if you said it was for the temp gauge to a Ford parts guy then you would, most auto parts guys wouldn't know what you needed if you just said "for engine temp", ECT Sensor would be first part they would hand you, not the sender
Hottest place on an engine is at the upper radiator hose which is why most temp senders are located there.
And the thermostat.
Your thermostat could be opening too much, spec temp is 192/195 thermostat.
But since sender is at the back of the engine and lower down gauge would be a few degrees cooler than actual temp.
If you want to change the thermostat just to see if that's the problem, then I would wait a bit to make sure there are no coolant leaks