I haven't run into this terminology before, but here is my guess. The term closed probably refers to the set of indices $n$ where the martingale is defined. That is, consider the domain
$$ \{ n : X_{n} \ \mathrm{is \ defined} \} = \mathbb{Z}_{\geq 0}, $$
which is not (topologically) closed in the extended reals because it has a hole on the right (at $n=+\infty$). Therefore, we define $X_{\infty}$ such that
$$ X_{n} = \mathbb{E}[X_{\infty} | \mathcal{B}_{n} ], $$
which is just what you call $X$. The domain then is closed.
From this, it should be clear that "closed on the left" is not a particularly useful concept, since the domain is always closed on the left (at $n=0$). I assume that for this reason the term "closed" is used instead, excluding the "on the right".
Notice also that essentially
$ X_{\infty} = \lim_{n \to \infty} X_{n}, $
although the actual nature of this convergence depends on what conditions we have on the $X_{n}$. See Doob's Martingale Convergence Theorem for this. This limit is useful in many applications such as filtering theory. Lastly, if we consider continuous time martigales $(X_{t} : t \in T)$, a more general notion of closedness may be useful, but in that case the left/right distinction is irrelevant since $T$ may well be a subset of $\mathbb{R}^{n}$ for $n > 1$.
