The sliding shorts by under armour are like the kind of underwear that look like super tight shorts and they have a built in pocket for the cup. My son says he doesn't notice it anymore but he would never be caught dead in a jock strap.
As for should you be wearing it....3 years ago the 2B on my sons team went after a hit ball that took a bad hop. It got him right in the crotch, he fell down and had to be carried off the field. That night his parents had to take him to the emergency room because one of his testicles was 3 times the size of the other one. They had to perform surgery to alleviate the pressure, it could have been avoided had he worn a cup. I have also heard of, but not been a witness to, a kid needing surgery to bring back down a testicle that had been hit so hard it went up, again, a cup would have saved him. At JV tryouts this year the 2B took a bad hop and needed help being walked off the field. No surgery needed but still, infield is wide open to these things happening.
Your mom is not over reacting. It might be only 1 in a thousand kids that get hit playing an infield position but she's trying to make sure that if it is you then you won't need surgery.