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The Bestvina-Brady construction of non-finitely presented groups of type FP produces groups of cohomological dimension two. Bestvina-Brady groups are parametrized by finite flag simplicial complexes. The Bestvina-Brady group is of type FP iff the flag complex is acyclic (=has the same ordinary homology as a point), is finitely generated iff the flag complex is connected, is finitely presented iff the flag complex is simply connected. For an acyclic flag complex the cohomological dimension of the Bestvina-Brady group is equal to the dimension of the flag complex.
So if you apply the Bestvina-Brady construction to any non-simply connected 2-dimensional acyclic complex you get a group having the properties that you asked for.
My `generalized Bestvina-Brady groups' allow you to construct uncountable families of groups of type FP, each having cohomological dimension two. The main novelty in my article with Tom Brown that you referenced in your question is that we use very different methods, whereas previous constructions all used CAT(0) cubical techniques.