Terminology sometimes differ between what field of mathematics you come from, or may even differ between mathematicians. In this case, you have found a concept which, depending on who you ask, is well defined proabably as your case 1 or case 2.
This is the reason why it is important in a good book to have clear definitions, even of the easy stuff, since it may differ between authors exactly what they mean when they say a concept.
Another example of this is the concept of subgraph. In graph theory it is common that a subgraph may exclude only edges (hence if you remove all edges from a graph, that is a subgraph of the original graph). While if you come from more the logic side of mathematics and talk about subgraphs, you probably mean that a subgraph is just taking a subset of the node set, and keeping all the edges (hence the graph with all edges removed isnt a subgraph if you had edges in the original graph).
The deeper into mathematics you go the more common this gets, since newer concepts often have less established names.
Conclusion: Its just terminology which isn't completely established. Empty graph and null graph are may both be either the graph without any vertices, or a graph with vertices but without any edges. It is hence important for evertone to briefly hint on what they mean when they state talk about the empty graph.