Knickers are undergarments and bloomers (almost archaic) are baggy knickers
Shorts are outer garments.
I have not heard any word other than shorts to describe 'short trousers' except in the specialised area of tight elastic bicycle shorts that are referred to as knicks. The difference with knicks is that they are skin tight and not really short in that they extend down to the knee.
What do they wear, then? Off topic, but in Japanese, we call it ブルマ(bloomer). It's not underwears here. The name and the habit of wearing it in PE (though it's almost died out because of hordes of horny men) comes from the eponym, Ms. Bloomer.
I'm talking about the panty-looking "shorts" NT volleyball players wear. Is it really "shorts"?
In English, Cheshire, "bloomers" (same derivation you mentioned) are no longer worn, except by a certain very tiny minority religious group. The only link I could find was to a commercial site specializing in clothing for these ladies. Since I believe it is against posting rules, I'll send it to you by PM so you can see what we're talking about.
In any case, as the Punctuation Man has correctly stated, in women's clothing "shorts" are outer wear, over panties. Bloomers, whether historical or for this small religious minority, are worn under long and ample dresses.
When I played netball at school, we were subjected to what were known as "knicker shorts" - they have the shape of knickers (underwear) but are of thicker material. They sound something like what you were describing - but in netball, we wore a skirt over them.
Their purpose, as far as I could tell, was to prevent the whole world (or the 2-3 supporters we used to pull in...) seeing your underwear when you were ruthlessly shoved to the ground during this supposedly "ladylike" sport!