fall into a category vs. fall under a category

Question

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?

Answer 1

In your example:

Notice that these examples fall under well-established categories of acceptable, standard English.

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:

Voters fall into three main categories.
Students over 25 fall into a different category.

Here, the voters (or students) are part of (inside) the mentioned category. Therefore, fall into is correct.

Answer 2

Falls under category is to be used when you already have referred to the names of the category. For example

We first divided voters into three categories, A, B, and C.

Voters aged less than 30 falls under the category, so ---

The literature reviewed in this chapter falls into three categories, viz...

Answer 3

It does not help anyhow to make a choice between falls under or falls into that the categories have been previously defined. You can use anyone of the expressions falls under the category or falls into the category.

And that hierarchical form and sublevels or spatial partition into different categories, it depends more upon how you conceive a concept that will decide for you which one to use. I think in most of the cases both can be used without compromising degree of accuracy.