In Wikipedia order topology is defined by the subbase consisting of $(a,\infty)$ and $(-\infty,b)$.
Why is not it defined by intervals $[a,\infty)$ and $(-\infty,b]$ instead? Is this Wikipedia definition (with open intervals) accepted by all or absolute most of mathematicians?
Consider an one-point ordered set (call this point $0$). With the Wikipedia definition we have an empty subbase. Isn't it better to have the subbase consisting of the set $\{0\}$?
If you define the intervals $[a,\infty)$ and $(-\infty,b]$ to be open, then you just get the discrete topology. Indeed, for any $a$ in your set, $\{a\}=[a,\infty)\cap(-\infty,a]$ would be open. On the other hand, defining it with open intervals gives the standard topology in the case of $\mathbb{R}$, which is pretty strong evidence that this definition is useful at least in some contexts.
There is nothing wrong with having an empty subbase. When generating a topology from a subbase, you can take finite intersections of your subbase sets. This includes the intersection of no subbase sets (the intersection of all the elements of the empty subset of the subbase), which gives you the entire space.