These still aren't the most general definitions for martingales. The most general definition I know of is:
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:
- $X_i$ is $F_i$-measurable for all $i \in I$
- 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.
For example, your quote from Kallenberg's book is only an introduction. He later defines martingales on the real numbers and only needs these types of martingales in the book. The book of Stroock and Varadhan mainly deals with stochastic analysis. Herein, the assumption of right-continuity is useful and a prerequisite for many theorems. They need the special assumptions, so they simply add them to the definition of martingales.