As far as I know, when a group of people or things with particular features in common are classified we can use "fall into a category". But quite recently I came across the following sentence in a linguistics book published by Hodder Education:
Notice that these examples fall under well-established categories of acceptable, standard English.
There is no record of "fall under a category" in none of my learner's dictionaries. Instead, they all say "fall into a category". For example:
"Voters fall into three main categories."
"Students over 25 fall into a different category."
So, is "fall into" different from "fall under" when we talk about categories?
Best Answer
In your example:
Think of a hierarchal chart with categories and sub-categories. In your example, "these examples" are sub-categories of "well-established categories", as far as I understand the context.
In such a chart, sub-categories are below categories, so fall under is correct.
In your second examples:
Here, the voters (or students) are part of (inside) the mentioned category. Therefore, fall into is correct.