It may be that my question is answered in Lovász's book:

Lovász, László. Graphs and Geometry. Vol. 65. American Mathematical Soc., 2019.

p.82:

Theorem 6.2. Every planar map in which the unbounded country is a quadrilateral, all other countries are triangles, and is not separated by a $3$-cycle or a $4$-cycle, can be represented as a resolved tangency graph of a square tiling of a rectangle.

Concerning corner touching, Lovász says:

We can specify arbitrarily one diametrically opposite pair as 'infinitesimally overlapping,' and connect the centers of these two square[s] but not the other two squares. We call this a resolved tangency graph of the family of squares.

Indeed the source is Schramm 1993.

My question is answered in Lovász's book:

Lovász, László. Graphs and Geometry. Vol. 65. American Mathematical Soc., 2019.

p.82:

Theorem 6.2. Every planar map in which the unbounded country is a quadrilateral, all other countries are triangles, and is not separated by a $3$-cycle or a $4$-cycle, can be represented as a resolved tangency graph of a square tiling of a rectangle.

Concerning corner touching, Lovász says that, in the case where four squares share a vertex,

we can specify arbitrarily one diametrically opposite pair as "infinitesimally overlapping," and connect the centers of these two square[s] but not the other two squares. We call this a resolved tangency graph of the family of squares.

Indeed the source is Schramm 1993.