Let $E$ be a Polish space and consider the Skorokhod $J_1$-topology on the space of cadlag functions $D_T := D([0, T], E)$ - this is just the "common" topology on this space as used in the theory of convergence of cadlag stochastic processes. This topology is metrizable and can therefore be completely characterized by the following notion of sequential convergence.
Let $\Lambda_T$ denote the set of continous strictly increasing functions $\lambda : [0,T] \to [0,T]$ satisfying $\lambda(0) = 0$ and $\lambda(T) = T$. In $(D_T, J_1)$, a sequence $x_n$ converges to $x$ if and only if there exists $\lambda_n \in \Lambda$ such that $\lambda_n \to id$ uniformly in $[0,T]$ and $x_n \circ \lambda_n \to x$ uniformly in $[0,T]$ (by choosing any compatible metric in $E$).
One can also define a $J_1$-topology on $D([0, \infty), E)$. But this is usually not performed in terms of a strictly categorical projective limit approach, simply because the restriction mappings $D_T \to D_{T'}$ for $T' < T$ are not continuous everywhere. The problem is that in $(D_T, J_1)$ the right endpoint $T$ of the interval $[0,T]$ plays a special role. For instance, if $E = \mathbb{R}$ and we set $x_n := \chi_{[0,T'-\frac{1}{n})}$ and $x := \chi_{[0,T')}$ where $\chi$ is the indicator function, then $x_n \to x$ in $(D_T, J_1)$ but $x_n$ does not converge in $(D_{T'}, J_1)$. The source of the problem seems to lie in the choice of the set $\Lambda_T$, since any $\lambda \in \Lambda_T$ is required to satisfy $\lambda(T) = T$ and to be continuous in $T$.
Now what happens if we drop the continuity condition of $\lambda$ at $T$? In other words, let us consider the set $\Lambda_T' \supseteq \Lambda_T$ consisting of all strictly increasing functions $\lambda : [0, T] \to [0, T]$ satisfying $\lambda(0) = 0$, $\lambda(T) = T$ and $\lambda$ is continuous on $[0, T)$, so that $\lambda$ may have a jump at $T$. Use this set $\Lambda_T'$ in place of $\Lambda_T$ to define a $J_1'$-topology on $D_T$. Then the restriction mappings are continuous and the family $(D_T, J_1')$ forms a projective system of topological spaces. Its projective limit is then just the usual $J_1$-topology on $D([0, \infty), S)$.
Question: What is so special about the $J_1$-topology when compared to the $J_1'$-topology? For me, it seems that $J_1'$ has better categorical properties and is also possibly simpler to work with, because the right end point $T$ of the interval $[0, T]$ does not need to be treated in a special way.
This is not quite true. Let $T = 1$, and let $x_n(t) = \mathbf{1}_{\{t \geq 1 + 2^{-n}\}}$. Then it's easily verified that $x_n \to x$ in $D([0,\infty),\mathbb{R})$, but $x_n[0,1] \not\to x[0,1]$ even in your $J'_1$ topology, since for all $\lambda \in \Lambda'_1$, $\lambda(1) = 1$ so $|x_n - x\circ\lambda| = 1$.