I am reading this paper.1 Near the middle of page 2 the author seems to state that a framed link is just a ribbon graph. Is that an accurate statement?
1Peter Tingley: A minus sign that used to annoy me but now I know why it is there (notes on the Jones polynomial).
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.