Suppose $K$ is a simplicial complex with base vertex $v_0$ and $p$ is an edge path in $K$ based at $v_0$ (so a sequence of vertices connected by edges starting and ending at $v_0$). In other words, $p$ is a simplicial map $p:M\to K$ where $M$ is some subdivision of $\partial\Delta_{1}$. If the loop $|p|:|\partial\Delta_{2}|\to |K|$ on realizations is null-homotopic is seems, according to this page, that there is a sequence of edge paths (where each step moves edge(s) across a 2-simplex) from $p$ to the constant edge path.
I would like to know if $|p|$ being null-homotopic also means there is a subdivision $N$ of the standard 2-simplex $\Delta_2$ so the subcomplex of $N$ corresponding to the boundary $\partial\Delta_{2}$ is precisely $M$ and a simplicial map $h:N\to K$ and $h$ restricted to $M$ is $p$? Perhaps this subdivision can be "built" out of the moves?
Of course, you can take a simplicial approximation to a homotopy $|\Delta_2|\to|K|$ that extends $|p|$, but what I am asking seems more combinatorial. If anyone can suggest texts/papers for me to look into that treat this question that would also be very nice!