Too much contemporary science writing falls under the category of 'infotainment' (The Guardian); This definitely falls under the category of “don’t try this at home.” (mprnews.org); A combination of disabilities (not including deaf-blindness) that restricts a child’s learning falls under the category of multiple disabilities. (cerebralpalsy.org)
Thank you so much.You can use them interchangeably. It just depends if the speaker imagines the category as something clearly delineated, something with a line all the way round it, in which case he will say fall into, or as a heading or the name of a list, in which case he may prefer to say fall under, but even with a list or a heading, you can still say fall into. I don't even think the speaker would make that distinction, in reality.
If you Google the respective phrases, I think you'll find them used without any difference in meaning. Certainly in the examples I quoted in #7, either version could have been used in all the cases.