MOSFET terminology seems to vary in every article I read
So what is it then?

This is the correct graph: -

Taken from this wiki page.
Saturation refers to the channel being saturated and, as you said, no matter what \$V_{DS}\$ you apply, current remains constant. It is sometimes also referred to as the active region (not to be confused with the MOSFET being activated or ON).
The linear (triode) or ohmic region is when the MOSFET is used as a switch (ON). Your top graph (in the question) is basically wrong because it doesn't correctly show the different slopes in the ohmic region when you apply different gate voltages. This region is called linear because there are different slopes that are governed by the gate voltage and means the MOSFET can act like a variable resistor hence, it gets the name linear but, different texts use this term rather loosely.

Also just to be clear is the reason we want to minimise time in the
saturation region as a FET because here we have a VDS and an ID
current flowing through the FET? Whilst as soon as the fet reaches
linear region, it's VDS would fall close to 0 and hence minimise
losses?

Yes, we want to minimize time in the saturation region because that is when the MOSFET is dissipating the most power and potentially operating below it's ZTC (zero temperature coefficient) and may suffer rapid thermal runaway.

Shouldn't the ohmic region be called saturated?

No, because in the ohmic region the channel isn't saturated. However, for a BJT, that equivalent part of the characteristic is called the saturation region but, for different reasons; for a BJT, it is the base that becomes saturated. Same name, different mechanism, different part of the characteristic.