On the definition of continuous time martingales Stroock Varadhan $\times$ Kallenberg

In the definition of martingales, one finds in Stroock and Varadhan (Multidimensional Diffusion processes - page 20) the strange request that it be right-continuous process.

However no such requirement is made in the wiki https://en.wikipedia.org/wiki/Martingale_%28probability_theory%29

nor in Kallenberg (Foundations of modern probability - page 96)

Isn't this strange of the part of Stroock and Varadhan? Has the notion of martingales evolved?

Let $I$ be a poset. A Family $F$ of sigma-algebras is said to be a filtration if for all $s, t \in I$ with $s \le t$ the inclusion $F_s \subset F_t$ holds. A family $X_i$ of (quasi)integrable random variables is called a martingale with respect to $F$, if:
1. $X_i$ is $F_i$-measurable for all $i \in I$
2. If $s \le t$, then $E[X_t | F_s] = X_s$
The problem with this definition: It is rarely necessary to use it. In most cases $I = \mathbb{N}$, $I = \mathbb{Z}$ or $I$ is an interval in $\mathbb{R}$. It is not uncommon for an author to omit the most general definition, if he only uses some special instances.