Graph terminology: vertex, node, edge, arc Precisely speaking, what is the difference between the graph terms of
("vertex" vs. "node") and ("edge" vs. "arc")?
I have read that "node" and "arc" should be used when the graph is
strictly a tree.
If there is a precise rule or protocol, please cite a reference.
 A: 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
graph.

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
makes sense.

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.
A: The distinction between vertex and node seems to me to be mostly about discipline (e.g. whether you come from combinatorics or computer science) and is irrelevant. The distinction between edge and arc can sometimes be relevant depending on who's using it: combinatorialists sometimes use "edge" to mean "undirected edge" and "arc" to mean "directed edge," although this usage is not universal. 
A: My readings suggests "arc" and "edge" are conceptually the same. Yet, LEMON ( http://lemon.cs.elte.hu/trac/lemon )has separate functions/methods for arcs & edges. I have played with it but not enough to understand the difference, in their usage. 
A: As I was wondering the same, I stumbled upon some extra information. I'm reading a book about Computational Geometry.
In this book a graph is drawn within a polygon, to avoid ambiguity they used edges to talk about parts of the shape and arcs to talk about the graph.
The book says the following

$\dots$ let $G$ be a graph associated with a traingulation, whose arcs are the edges of the polygon $\dots$

A: In my opinion, both terminologies come from the different perception of graphs and networks. Albert-László Barabási writes in his recent text book.
"In the scientific literature the terms network and graph are used interchangeably:




Network science
Graph theory




Network
Graph


Node
Vertex


Link
Edge




Yet, there is a subtle distinction between the two terminologies: the {network, node, link} combination often refers to real systems: The WWW is a network of web documents linked by URLs; society is a network of individuals linked by family, friendship or professional ties; the metabolic network is the sum of all chemical reactions that take place in a cell. In contrast, we use the terms {graph, vertex, edge} when we discuss the mathematical representation of these networks: We talk about the web graph, the social graph (a term made popular by Facebook), or the metabolic graph. Yet, this distinction is rarely made, so these two terminologies are often synonyms of each other."
<tl;dr> Same, same, but different.
