Can pseudochiral be called as a subgroup of chiral?
No. Chirality is a property of a molecule as a whole. A molecule is either chiral or achiral.
For a molecule to be chiral, it must have a non-superimposable mirror image - that is the definition.
Pseudoasymmetric is a property of a point - i.e. you can use it to describe a centre, but it is not descriptive of the molecule as a whole.
A molecule can have multiple asymmetric centres (a.k.a. chiral centres) and still be achiral (see:
meso compounds) and some molecules are chiral despite having no asymmetric centres (see: axial chirality, for example). This is why I personally think the term "chiral centre" should not be used.
Asymmetric centres need not be at C atoms or involve sp
3 hybridisation. There are plenty of chiral metal complexes where the asymmetric centre is the metal (e.g. Ru(bipy)
3), and chiral phosphines and sulfoxides with asymmetric centres at P and S feature in many ligands and auxiliaries used in asymmetric synthesis.
The two cyclobutane structures you posted are both achiral.