I see several answers to your questions.
I was able to retrieve the original citation "A 2.4-GHz 0.18-um CMOS Self-Biased Cascode Power Amplifier". The basic problem is that you have cut the "conventional" and "self-biased cascode" schematics from a paper without explaining the discussed circuit purpose and the relation of both derived by the author.
To give a literal answer to your question, M2 isn't operated in a classical common-gate circuit in the self-biased cascode in post #1, but it's still near to common gate. In my view it's pointless to discuss details without referring to the circuit purpose given in the paper.
By the way, what's your motivation to analyze this specific circuit? Are you doing any work related to RF IC design in sub-µm technology, or is it just a misunderstanding?