The article is also using an extremely broad definition of the labor force: anyone over the age of 16. That definition can be useful for some demographic calculations, but I don't think it supports the conclusions being drawn here. In addition to your categories, it includes millions of retirees.

What on earth is wrong with that? Just because you're still in high school is no excuse not to be pulling your weight 40 hours a week (harrumph)

Also, the article starts off saying "The total non institutional civilian labor force (Americans 16 years and older who are not in a institution -criminal, mental, or other types of facilities- or an active military duty)", then goes on to discuss further the breakdown without ever mentioning military employment again.

When it comes to the military, you're only talking a couple million people; Not significant enough to move the needle.

"# Part time employed for non-economic reasons: 18.184 million people. Non-economic reasons include school or training, retirement or Social Security limits on earnings, but also childcare problems and family or personal obligations."

I would assume those people would fall into that category, but I might be missing something.

That category is those who are part-time employed due to these reasons. People who have no job at all because of these reasons – perhaps because they don't even want a part-time job – fall into the 100+ million people over 16 neither working nor seeking work.