Here is a post with some measurements about how the temp sensor in the lower radiator hose relates to the ECU turning on the aux fan.
https://xoutpost.com/1210848-post45.html
For some reason, I made it a little research project to separate fact from folk lore regarding turning the fan on (while debugging my fan that would not spin; eventually fixed when I replaced the fan), and started trying to measure everything rather than believing youtube or the intertubes.
In that post there, I measured (my car is a 2001 3.0i) what the DME (as reported by Foxwell) thought the lower radiator hose temp was as I swapped in different resistors (so the actual sensor was unplugged from the connector).
I did confirm that the 100 Ohm resistor simulated a 241*F temperature, which triggered the fan to come on. So from that, you can conclude that - no surprise - the fan will be turned on when the ECU thinks the lower radiator hose is at 241*F, regardless of other factors.
With a failed fan, I could tell it was turned on by measuring the signal the ECU sends to the fan - it varies the amplitude of a ~1Hz square wave.
In my case, I was just looking to confirm that it would trigger, and get an idea of the resistance. I was not looking to find the exact trigger point.
So I'll suggest that it's a pretty easy experiment to do - either with a potentiometer (in the range of 100-200 Ohms should do it) or a bunch of individual resistors - set whatever operating conditions you want (AC on/off, engine idling, whatever) and see what triggers the fan. Once you know that resistance, if you don't have a Foxwell, just let me know, and I'll measure it on mine for you.
You may find a variable speed output as the temperature ramps up. May be tough to tell exactly by looking at the fan, but since I was looking at the control signal, I was able to see this vary precisely.
I have a few other posts on this site with details about the fan testing I did. My main goal was to confirm the fan was actually bad before replacing it, and then due to all the inaccurate folklore, I became curious about how things actually worked.
EDIT - also, re-reading your first post here as well as the following two posts below this one, some basics ...
The thing I looked at here was how the ECU triggers the AUX fan based on the lower radiator hose temp sensor. That sensor is not the one used for the instrument panel gauge, and possibly other engine cooling control functions. For example, I believe the cylinder head temp sensor is on the left side of the head, between cylinders 4+5, and is used by the ECU to control the engine cooling thermostat (sending current to open the thermostat wider than it would by coolant temp alone).
On @
AndrewWynn's following post about rigging something up to adjust the setpoint, if you can confirm what temp the fan turns on at, and what the resistance of the sensor is at that point, and what the resistance would be at the temp you want the fan to turn on at, you could permanently add a new resistor in parallel with the sensor - that would make the ECU think the lower radiator hose is hotter than it really is, turning on the aux fan at a lower actual temperature. But if the coolant thermostat is still doing its thing, this might not be a great way to manage engine temp.
If you just want to turn the aux fan on during vacuum leak debugging, there are easier ways. Off the top of my head, if you turn on AC and unplug the AC line pressure sensor, that will do it within seconds. I'm sure I've posted details on this somewhere here.