That is nearly accurate. A ribbon graph is usually an inclusion $\Gamma\to\Sigma$ that is a homotopy equivalence between a graph $\Gamma$ and an oriented surface $\Sigma$ with boundary (one of the many ways to define a _combinatorial map_ ). A ribbon graph embedded in $S^3$ is called various things, such as a _spatial graph_ or a _spatial ribbon graph_ , diagrams for which have been called _flat vertex graphs up to regular isotopy_. A framed link is such a spatial graph where every vertex is degree-$2$, modulo edge subdivision. The correspondence is that the surface $\Sigma$ gives a section of the normal bundle of the embedding of $\Gamma$ in $S^3$.
The $\mathcal{RIBBON}$ category later in the paper is a category of framed tangles. Or, the category of oriented spatial graphs such that interior vertices are all degree-$2$, modulo edge subdivision.
If it were just of ribbon graphs, then there'd be no concept of over- vs. under-crossings, just permutations.