What is a "proper face" of a graph? This document uses the term "proper face". I do not know what this term means. At a guess, perhaps it could mean "a face that is not the external face"; but since the term "internal face" is a more obvious way to say the same thing, this seems somewhat unlikely.
I looked at the Wikipedia page "Glossary of Graph Theory", but it does not define this term (or at least, the string "proper face" does not occur anywhere on the page). I also searched this Stack Exchange site for the term, but the search did not return any results that I could understand and that related (so far as I can tell) to graphs.
 A: You may rest assured that:

*

*This is not common terminology, and

*Neither of the two textbooks cited in "We assume the reader is familiar with basic graph theory appearing for example in [11, 26]" ever uses the term.

I would guess that "proper face" means "internal face" even though the term "internal face" would make much more sense. This is at least plausibly consistent with the context in the paper.
I did find another paper, An algorithm for the graph crossing number problem by Julia Chuzhoy, that seems to use "proper face" in this way. That paper defines $F_{\textsf{out}}$ to be the "outer face" and twice says that something is a "proper face, distinct from $F_{\textsf{out}}$".

Some related terminology that doesn't help:

*

*A proper face-coloring of a planar graph is a coloring of the faces such that adjacent faces get different colorings. It has nothing to do with "proper faces": it is a "proper coloring" of the faces.

*In the context of simplices or polytopes, a face doesn't have to be two-dimensional; vertices, edges, and higher-dimensional analogs are also considered faces. In this case, a proper face of $P$ is just a face that's not all of $P$ (by analogy with proper subsets).

