It's all kinds of confusing.

AFAIK there is no botanical notion of vegetable. It's a culinary/food term, which should just mean "edible plant matter". Which would make culinary fruit just a subset of vegetables. But for some reason popular usage dictates that the category "vegetable" excludes some sweet tasting fruit, staple starches and some other things.

Seedless watermelons still have seeds, they just aren't mature when the fruit is ripe. This means they are smaller and are soft enough that eating them does not give crunch.

You’re correct, the author of the article was confused. Probably got confused because watermelon are related to crops that are culturally considered to be vegetables, like cucumbers. Another thread mentioned the same.

Yes, it looks like they're fruit botanically and culinarily, but they're vegetables "agriculturally", whatever that means.

It is the fruit we eat. However, the fruit/vegetable distinction does not make a lot of sense botanically.