There's obviously a misunderstanding of technical terms. In MOS literature, triode region is defined by the boundary condition VDS ≤ VGS - VTH, while in saturation (or in traditional vacuum electronics terms pentode) region VDS > VGS - VTH applies.
The region near the origin with resistive VDS versus ID behaviour is also named deep triode region. Of course, the latter should be reached in switch on-state. So far regarding definition of terms.
I guess, that part of the confusion comes from the fact, that a BJT power switch is usually operated in saturated (IB > IBmin) mode, not related to saturation of MOSFETs anyhow.
But apart from the meaning of terms, it's clear that static switch losses can be only minimized by reducing VDS at a given ID. You don't need a region definition from MOSFET theory to understand this simple fact.