I guess generally,they are v.close....
the problem is you need to get a point where your switching of liner to sat. is happening and accurate W/L for that may not be applicable in spice/layout.
Also vdsat doesn't always remain vgs-vth due to lamda effect..., it starts increasing linearly!
In hspice simulation result report file , u can see vdssat and vgs .vds, Vdssat =Vgs-vth , but vds is the real value . For mos working in saturate region , vds must
greater than vdssat
I could actual not refer to the equations but the vdsat is defined from the strong inversion to the subthreshold range. The first model equation known to me was the EKV model which use a single equation for both ranges. There was also the defintion of the vdsat.
In subthreshold the lower limit is Vdsat=m*Vt where m is the subthreshold slope.