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.
When you pull the slingshot back on a catapult, you are using muscular force from your arms and hands to stretch the elastic material of the slingshot. This builds up potential energy in the slingshot, which is then released when the projectile is launched.
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.