Based on the comments, I missed one case:
1. Add an unconnected node.
I considered another approach to answering the same question. Consider the set of edges E in the original graph, and the set E' after adding the new node. How do these differ?
The missed case was the edge case E=E'. If the two sets are not equal, there are either edges in E that are not in E', edges in E' that are not in E, or both.
The "no unrelated changes" requirement means that any new edge starts or ends in the new node. It also means that an edge which existed in E but not in E' was not simply removed, but changed, and therefore there must also exist a corresponding new edge in E'.
Hence, we have two possible explanations for new edges in E', but only one explanation for edges that have gone missing from E. So if N is the number of new edges in E', and M is the number of missing edges from E, then we know N>=M. We can match M out of the N new edges to the M missing edges; each of these M pairs of (old edge, new edge) represents a "changed" edge. 835
(This might not be a unique pairing). This will leave N-M edges that are entirely new.
We can split the M changed edges further in two groups; those edges that were changed to point to the new node and those that were changed to originate at the new node. (per the "no unrelated changes" assumption). Similarly, we can divide the new edges in two groups.
At this point, we have found back the categories 1-4 from the original answer. As noted in the comments, the 5th category (spliced in) can be represented as a combination of changes 1+4 or as the combination 2+3. It can make sense to semantically identify this as a unique change, because splicing in one node keeps the graph connected (and it remains a DAG). This gives us the full set of 6 change types that were identified before.
I have intentionally ignored adding edges that both originate and terminate at the new node, because that type of change certainly introduces a cycle. No other single change will introduce a cycle, but it's fairly obvious that if the old DAG had at least one edge, you can trivially add a new node and two new edges to introduce a cycle containing that original edge. 242