A tree gives rise to a simplicial 1-complex. A tree like simplicial 2-complex would be simplicial 2-complex without any closed 2-subcomplexes (the analog of a cycle in graphs) and such that the 1-skeleton given by the subcomplex where more then 2 cells meet has no cycles (forms a tree).
I wonder how to prove that the following graph (of the free modular lattice on 3 generators) cannot be embedded into such a simplicial 2-complex.
Has this generalization of planar graphs been characterized? Does it have an excluded minor characterization at all?
I think that it can be embedded. Consider a 2-complex obtained by adding the edge in the middle and then the obvious 2-faces. Concretely, it means adding 3 triangles and $3\cdot3$ + $2\cdot6$ quadrilaterals. It is not acyclic yet, but after removing one quadrilateral from each cube, you get an acyclic one. Now you may subdivide the remaining quadrilaterals in order to obtain a simplicial complex.
Note: I interpret your last condition that in your 1-skeleton you consider the edges which are in more than two 2-cells. (That is, a triangulation of a plane would be OK except finiteness.)