What is the correct terminology for the following property of a simplicial set $X_\bullet$:
For every $k\geq 0$, every map $\partial\Delta^k\to X_\bullet$ can be extended to a map $\Delta^k\to X_\bullet$.
This is just a contractible Kan complex. It's equivalent to the same condition where you replace the pairs $(\Delta^k,\partial\Delta^k)$ with all pairs $(A,B)$ where $B\subset A$, since $A$ can be built from $B$ by iteratively filling in simplices whose boundaries are already filled in. In particular, any such $X$ will satisfy the Kan condition, and applying the condition to the pair $(X\times \Delta^1, X\times\partial\Delta^1)$ gives a contraction of $X$.
Conversely, given a contractible Kan complex $X$ and a map $\partial \Delta^k\to X$, by contractibility it extends over the cone on $\partial\Delta^k$. But that cone is just a $(k+1)$-horn, and filling in the horn gives a $k$-simplex in $X$ extending the original map.