A stronger Flow Decomposition Theorem?
The statement of Ahuja, Magnanti, and Orlin applies to general flows, not only $(s,t)$-flows. During decomposition, when removing a path flow, either one of the two endpoints become balanced or one the arcs will get flow 0. This gives the bound $m+n$.
In the case of $(s,t)$-flows, we may note that during decomposition the nodes $s$ and $t$ becomes balanced at the same time. This gives the already the bound $m+1$ on the number of paths and cycles. If we look at the flow decomposition, when finding the final path flow (this is where the balance of $s$ and $t$ becomes balanced), if no flow on some arc is made 0, it means that we will still find a cycle flow. Now the final cycle flow will make at least two arcs have flow 0 at the same time. This sharpens the bound to $m$ in case of $(s,t)$-flows.
P.S. The proofs of the notes you link are not quite correct, see the comments.