Does the heap property in the definition of binary heaps apply recursively?
I would think that most people would just say that the heap property applies only to a vertex and its immediate children, because it's a sufficient definition to maintain the "recursive" heap property everywhere.  It's a cleaner way to think about it (you only need to check O(1) objects per vertex to verify that a tree is a heap, as opposed to O(n) per vertex).

I don't know if there is exactly a consensus on this, and perhaps some other people think differently.