The way I learned it, a monotone class on a set $X \neq \varnothing$ is a collection of subsets of $X$ that is closed under monotone countable unions and intersections. According to this, $X$ is not necessarily an element of $M$.
However, Wikipedia says that $X$ must be an element of $M$. So,
1) Is it more common to regard $X$ as an element of $M$?
2) In practice, does it matter whether or not $X \in M$?
I am asking 2) because I only used the concept of a monotone class in two situations and I do not remember having to worry about whether or not $X \in M$.
The Wiki definition here says the overall set $X$ is in any monotone class $M$, and also any countable union/intersection of nested sets from $M$ is in $M.$
If the definition did not explicitly postulate that $X$ is one of the sets in $M,$ then the empty class of sets would be a monotone class. Maybe one doesn't want that...