Why is the Triode operating region of a MOSFET named such?
Why is the Triode operating region of a MOSFET named such?

When a FET is operating in the triode (ohmic) region, the drain current is strongly dependent on drain-source voltage.  The plate current of a vacuum tube triode is also strongly dependent on plate-cathode voltage and this is, I believe, why the triode region of operation is so named.
FET drain curves operating in triode region:

Vacuum tube triode plate curves:

This strong dependence of drain (plate) current on \$v_{DS}\$ (\$v_{PK}\$) severely limits the available voltage gain of the device.  The greater the dependence, the less available gain as this dependence is, in effect, acting as a resistor in parallel with the load, shunting signal current around the load.
So, although triodes were (and still are) used as amplifiers, tetrodes and pentodes were developed that, like the saturation region for FETs, greatly reduce the plate current dependence on plate-cathode voltage.  These devices offer, among other things, much higher amplification factors, i.e., they have much higher plate impedances.