What is this binomial sum?

Let’s look at it a little differently. Suppose that you want to choose a set of $m-n+2$ numbers from the set $A=\{0,1,\ldots,m\}$. If $S$ is such a set, let $k_S$ be the second-smallest member of $S$. Then $S$ has $1$ member smaller than $k_S$ and $m-n$ members larger than $k_S$. For a given $k$ there are $k$ ways to pick one smaller member of the set $A$ and $\binom{m-k}{m-n}$ ways to pick $m-n$ larger members of $A$, so there are

$$k\binom{m-k}{m-n}$$

ways to choose $S$ with $k_S=k$. Summing over the possible values of $k$ gives the total number of such subsets, which is of course

$$\binom{|A|}{m-n+2}=\binom{m+1}{m-n+2}\;.$$

The problem with your approach is that when you choose one of the first $k$ and $m-n$ of the last $n-k$ elements of $[m]$, you’re not just making a selection of $m-n+1$ elements of $[m]$: you’re also specifying a break-point between the first one and the last $m-n$.