Longest path of a DAG containing a single node?
You have not misunderstood anything, there is simply a discrepancy of notation: sometimes the length of a path is measured as its number of edges, sometimes as its number of nodes, depending on context. In your implementation, it doesn't matter which convention you use, as long as you are consistent throughout your algorithm. Then if you use the second convention and your algorithm returns $3$ for a particular vertex, it means the longest path to that vertex has three nodes: $abv$. If you use the first convention (which is what this Wikipedia article uses, but not your professor), the algorithm should return $2$, because the longest path has two edges.

If this is an assignment, and your professor uses the convention that paths are measured by their vertices, and says that you should too, then you should too.