On complexity class $\mathsf{\Pi_2 L}$ I suggest the following definition of $\mathsf{\Pi_2 L}$  (similarly to the certificate definition of $\mathsf{NL}$):

A language $L$ belongs to $\mathsf{\Pi_2 L}$ iff there exists a deterministic Turing Machine $M$ and polynomials $p$ and $q$ such that 

$$x \in L \Leftrightarrow \forall z \in \{0,1\}^{p(|x|)} \exists y \in \{0,1\}^{q(|x|)} M(x, z, y) =1.$$
Here $x$ is placed on its input tape and $z$ and $y$ are placed
on its special read-once tapes, and $M$ uses at most $O(\log |x|)$ space on its read/write tapes for every
input $x$. We can read $z$ and $y$ simultaneously.

What can we say about this complexity class? Since $\mathsf{NL} \subseteq \mathsf{P}$ 
we have  $\mathsf{\Pi_2 L} \subseteq \mathsf{coNP}$. Can we say something else about this class? An example of a language from $\mathsf{\Pi_2 L}$ is here: Is unary $\Pi_2$-SUBSETSUM coNP-complete?
If I got this right, $\Pi_2L$ as defined above is equal to co-NP. The co-NP-complete DNF tautology sits in $\Pi_2L$: Use z for the assignment and y to choose which clause to check. To show $\Pi_2L$ in co-NP, universally guess z and the rest can be computed in P.