Saturation regions for BJTs and MOSFETs
why the saturation region of the MOSFETs has this name?


The post you link to does explain this, but in case it needs repeating and perhaps backing up with a textbook reference, the saturation region for a MOSFET is called so because the drain current saturates, i.e. basically stops increasing as Vds increases further.

You are correct that the active region of a BJT corresponds to the saturation region of a MOSFET when these devices are used as amplifiers.

The saturation region of a BJT (e.g. when turned on as a switch) corresponds to the triode/ohmic region of a MOSFET.

Some authors also call the saturation region of a MOSFET the "active mode", which does match the terminology used for BJTs. But they also call the triode/ohmic region the "linear mode" which perhaps doesn't help that much because "linear" suggests an amplifier rather than a switch. Linear here again refers to how the MOSFET characteristic looks like in that region rather than any external/use considerations. (Luckily, it seems nobody calls the BJT saturation region "linear mode".)

The only thing that's not confusing about this terminology is the cut-off region, which is the same for both. Here's a summary table for the correspondence (from an external/use viewpoint):



This summary also includes the reverse active region for BJTs, which is seldom used, but it doesn't include synonyms for the triode region; as I said "linear mode" or "ohmic region" are also used to denote the MOSFET triode region.