Definition for MSO2 for arbitrary structures
If I understand correctly, you just want a definition of the logic; then you can find a definition of MSO$_j$ (where 2 is a special case) by example in ''Elements of Finite Model Theory'' of Leonid Libkin.
Even if you are interested in infinite model, the definition of the logic is the same.

Quickly is just the set of formulae with existantial quantification over sets of the elements of the universe, then existantial quantifications over those sets; then a first-order formula.