You've got it!
The Vt is as you already know - the trigger voltage - so good to know the tester helps to develop a profile of the part BEFORE you try and put it in. At least you are aware of the variances - which is why I mentioned that "matching part".
The problem with "trigger" is many get confused with the performance of the part related to it's trigger - which helps you install the part, but does not give you a clue as to how the gate capacitance will affect how the part performs as the replacement.
Trigger is a relative voltage reference - that refers to how the device appears at that voltage it tested on as.
As you found out, the OEM part is/was a 3.7 which the replacement showed 4.1Vt is pretty close to 1/2V ABOVE the OEM one - so then you have to look at the power input to the Gate - as the "trigger level" - but not as a tuning or stability reference - which gate capacitance then becomes that figure in which to select from.
So I would have skipped the 4.1Vt and tried to locate another with Vt closer to the OEM - but that does not rule out the capacitance.
You may never find an exact match using Vt - but the paring - which was the question I see - relies on the balance of capacitance the OEM had to the one you're trying to put in there.
In Push-Pull or even single-ended supplies - Vt tells you it may need more power to stay linear, but Cg or the terminal capacitance it appears as, will determine the stability in there as a replacement.
If I may add a little more...
The question of the tester - the voltage is truly only a relative reference. There is the usual issues of how you zero out a device so sensitive that even the oil on your fingers and the flux in the wetted solder - or the oxides of a preservative to keep the part from rusting (reduction) - affects the trigger - due to the high impedance state the device you test - is working in - dirty tester claw or teeth - or even soldered on - doesn't matter - consistency in results does.
As you can see - even in the Datasheets, there is a variance in the dies results - so the burden of proof then becomes not what the device appears as - more in, how you test the device as well as the consistent results of that method of testing - provide. That gives you the predictability.
It's when you test a different Brand to another where you may find exceptional differences in results - but that may be from variables that if you had not tested using a method - you may never truly know how well the part will perform - until it is installed.
- Just so you know - the GATES are usually 2 to 4V - so the tester is already maxing out at 4.1V to even know the device is there.
- That can send up a red flag to me on why is it maxed out as ABOVE a typical ON result in the Datasheets?
- Might have more to worry about - but then too, the Oxides of March...keep marching on
This is why you don't see many aftermarket or such service parts as support for anything more than that of a simple part. "This device is not meant to be used in Life Support function"