Is there a more common term than "monotone" that describes this property? In A Topological Approach to Evasiveness they say:

A monotone is one which is not destroyed by the addition of edges.

Now if you look at the Wikipedia article on Graph Properties (particularly the section called "Properties of Properties") they define monotone as:

A graph property is monotone if every subgraph of a graph with
  property $P$ also has property $P$. For instance, being a bipartite graph
  or being a triangle-free graph is monotone. Every monotone property is
  hereditary, but not necessarily vice versa; for instance, subgraphs of
  chordal graphs are not necessarily chordal, so being a chordal graph
  is not monotone.

In a bipartite graph, the vertices can be broken up into two disjoint subsets $U$, $V$ and every edge in that graph connects a $u \in U$ and a $v \in V$. Take any two vertices in $V$, add an edge and you have destroyed the bipartite-ness. So the two definitions of "monotone" are not equivalent. I am going to assume that Wikipedia is using a more common usage of monotone, is there an alternative name for the other property?
 A: The main difference I see is that one definition considers adding edges, and the other considers removing edges. These are different but closely related. To disambiguate between the two, we typically talk about monotone increasing and monotone decreasing properties, respectively. In the context of property testing, we usually don't care about the distinction, since the negation of a monotone decreasing property is monotone increasing. But the distinction is there to be made.
The definition you found on Wikipedia seems to also allow you to take subgraphs with fewer vertices, whereas usually you just consider subsets (and supersets) of the edge set and keep the vertex set fixed. This will give you a slightly different notion in some unusual cases, and I don't know of any way to disambiguate between the two, but I think the Wikipedia definition is not the interesting version.
For example, I think most people would say that the property of having average degree less than $k$ is monotone decreasing, for practical reasons: the FKG inequality applies to it, it has a sharp threshold in the Erdős–Rényi model of random graphs, and it's evasive as Karp's conjecture would suggest. But a graph with average degree less than $k$ may have a subgraph with a higher average degree, so it is not monotone decreasing if you allow the removal of vertices.
A: The only graph theory research I have done is about the Sensitivity Conjecture and related problems, and in that subfield of graph theory the first is taken to be the definition of monotone as can be seen here. I would call the second definition "hereditary." I don't support using the term "monotonic decreasing" for the second property because of the issue with deleted vertices that Misha raises. Here are two important papers that use the first definition.

Noam Nisan. CREW PRAMS and decision trees. SIAM J. Comput. 20 (1991), no. 6, 999-1070
Wegener, L. The critical complexity of all (monotone) Boolean functions and monotone graph properties. Information and Control, 67:212-222, 1985. 

The only important result that comes to mind using the other definition is this paper but the proof appears to go through using either definition. I believe that Nisan's proof doesn't work for the second definition. At this MO thread the distinction is discussed in the comments.
