Yes, it is a type of torsion catapult, invented in 1888, used by children in the 1940's.
In Scotland, a slingshot is commonly referred to as a "catey."
The word you're looking for is.... catapult.
There are more but these are just a few: Aircraft Catapult, Slingshot Catapult, and an Onager Catapult.
Arbalest, slingshot, hurler, shooter, propeller...
I think what you are asking about was the trebuchet, which was much more like a sling than like a slingshot. There was a type of catapult called a ballista, which was like a large crossbow, and it is possibly a similarity to this type of weapon that is the reason people in England and many other places call a slingshot a catapult.
It was originally a catapult that they made smaller and they created it as the first weapon that they used when they were having war.
A catapult is a type of siege engine, sort of a non-chemical artillery. They hurled heavy spears by releasing the energy stored in torsion. In Britain, a "catapult" is an elastic toy that throws marbles or small stones, in the US the same thing is called a slingshot.
They typically have wings. Unless they've been launched from a slingshot or catapult, in which case wings are optional.
In most places, firing a catapult, or slingshot, is not a crime. However, some cities and towns have ordinances that can result in fines. Any damage done with one would be considered a crime.
Slingshot? Catapult? Trebuchet? Gun? Something that launches something else is usually good enough.
A catapult works because the gears and rope create a winch mechanism that winds up tightly. When the rope is released it launches an object into the air preferably at something.
There is no such item. But there is a slingshot. You can buy a sling shot in your Nook shop. If it isn't in your Nook shop today, look the day after, or the day after that.