What is serpentinization, in the context of disappearance of surface water on Mars?
does serpentinization just refer to the formation of some hydrated minerals that happen to be of a class that is historically been referred to as serpentinite or it's subgroup

Yes. In simple terms, the reaction is enstatite + forsterite + water = serpentine.
MgSiO3 + Mg2SiO4 + 2H2O = Mg3Si2O5(OH)4
Forsterite and enstatite (or more generally, olivine and pyroxene) are common minerals in mafic igneous rocks. The formation of serpentine locks up water in the resulting solid crystal structure.
You can see more examples of serpentine forming reactions on page 5 of this PDF file.

or is serpentinization a process that always produces serpentine patterns?

No. There is no such thing as &quot;serpentine patterns&quot;. The image you link to is of &quot;folded serpentinite&quot;. The &quot;serpentinite&quot; bit of it are the green layers, regardless of whether they are folded or  not. The nice photogenic folded part is a later tectonic event that is unrelated to the formation of the serpentinite.