You can take a look at "Introduction to Graph Theory" of Douglas B. West.
At page 3/Example 1.1.5 of the Second Edition:
The terms "vertex" and "edge" arise from solid geometry. A cube has
vertices and edges, and these form the vertex set and edge set of a
At page 55/Remark 1.4.8 of the Second Edition:
We often use the same names for corresponding concepts in the graph
and digraph models. Many authors replace "vertex" and "edge" with
"node" and "arc" to discuss digraphs, but this obscures the analogies.
Some results have the same statements and proofs; it would be wasteful
to repeat them just to change terminology (especially in Chapter 4).
Also, a graph G can be modeled using a digraph D in which each edge uv
E E(G) is replaced with UV, vu E E(D). In this way, results about
digraphs can be applied to graphs. Since the notion of "edge" in
digraphs extends the notion of "edge" in graphs, using the same name
So according to this book, vertices and edges are for undirected graphs due to the analogy with solid geometry and node and arcs are for directed graphs.
In computer science, "node" and "edge" are used in both cases.
I think the only mistake you could make is to use the term "arc" for undirected graph.