Vt is defined in datasheets as Vgs(th) = Vds at very low Id(th). (e.g. 100 uA) Vgs will be this value if Id <= Id(th)
I call it the "almost off" condition because it is specified at low currents often <= 0.1% of I max but never 0 uA when Vgs=Vt.
Does it makes sense to test at a measureable small current for Vgs(th) rather than the threshold at 0 current for the Vgs(th)?
Like hFE , in BJT's Vgs(th) has a wide tolerance ratio max/min >=2 But for this low current and lower Vgs=Vd is constant while being just slightly above Vt.
Thus using in saturation mode Vgs will always be constant and equal to Vt at some low current, whatever that is, which is also temperature dependent.
Did you skip this course?
Grok could write this better I'm sure.
You should learn to
simulate fast simple concepts My tool is the fastest. Use the one you know how to use fast. But I add real parasitics, ESR, ESL to make it more realistic when using high speed. Continue using LTspice or better ones for complex simulations.
The FET edit has two parameters Vt and beta which is like RdsOn but not quite with a link to choose it for any part but does not include C which is part of every PN junction. All parts are ideal in some sense. Like two caps together will cause infinite current unless you add ESR. Diodes and Transistors have physics models.
In the saturation region only,
Ids = beta/2 * (Vgs - Vt)^2
Comparing this with yours above, now you know what beta is used for.
Like Simplis you can put sliders on the side for any parameter for almost for any part
Grok can answer any simple question. Just be careful not to overlook some things in the prompts.
The better the question... the better the answer ™ (what you know, don't know and need to know, how reliable is it? how sensitive is it?)