What does it mean for a vector field to be "along" $\partial M$? I think "along" is a generalization of "on".

Another way to think about the distinction between a vector field $\mathit{on}$ $\partial M$ and a vector field $\mathit{along}$ $\partial M$ is as follows. A vector field $\mathit{on}$ $\partial M$ is a section of the tangent bundle $T(\partial M)$ of $\partial M$. A vector field $\mathit{along}$ $M$ is a section of $TM|_{\partial M}$, the restriction of the tangent bundle $TM$ to $\partial M$.

I think that choice 1 is correct: A vector field on $\partial M$ assigns to each point $p \in \partial M$ a vector $X_p \in T_p(\partial M)$. A vector field along $\partial M$, on the other hand, assigns to each point $p \in \partial M$ a vector $X_p \in T_p M$. That's different. The key phrase is "as opposed to $T_p(\partial M)$". The domain of a vector field along $\partial M$ is $\partial M$.