I have the following text from an academic course I took a while ago about inorder traversal (they also call it pancaking) of a binary tree (not BST):
If you read carefully you see that the first "definition" says to start left of the root and that the order of the nodes is determined by when you pass under them. So B is not the first node, as you pass it from the left on the way to A, then first pass under A after which you go up and pass under B. Therefore it seems that both definitions give the same result.