Parroting the link below (I imagine this is a standard definition), a topological space $T$ is $k$-connected ($k ≥ 0$) if, for every $0 ≤ r ≤ k$, every continuous map $f : S^r → T$ extends to a continuous map $f : B^{r+1} → T$. ($B^r$ is the $r$-dimensional unit ball.) A simplicial complex is $k$-connected if its geometric realization is.
http://web.cs.elte.hu/~lovasz/kurzusok/topol13.pdf#page=2
Parroting the link below, two simplices $\sigma$ and $\tau$ are $k$-connected if there is a sequence of simplices $\sigma$; $\sigma_1$; $\sigma_2$; $\dots$; $\sigma_n$; $ \tau$; such that any two consecutive simplices have at least $k+1$ vertices in common. A simplicial complex is $k$-connected if any two simplices of dimension greater than or equal to $k$ are $k$-connected.
https://math.la.asu.edu/~helene/papers/atheory_final.pdf#page=2
Are the above two definitions equivalent? If not, what is a more combinatorial way of describing what the first definition means?
They appear not to be equivalent. Take two tetrahedra (full complexes on four vertices) and join at a point. This gives us a simplicial complex on $7$ vertices whose geometric realisation is contractible.
By your first definition, this simplicial complex is $2$-connected. By your second, it appears not to be. It's not even $1$-connected.
A more combinatorial way of describing the first definition might be to say something like:
Two simplices $\sigma$ and $\tau$ are $k$-connected if there is a sequence of simplices $\sigma,...,\tau$ such that any two consecutive simplices are joined by a simplex.
However, I wouldn't go just altering your definitions without very strong motivation - whilst I know little about the topic of the second paper, there is sure to be good reasoning behind their definition.