an Onager catapult is a type of Mangonel catapult.

If that statement is correct then all onagers are mangonels.

But not all mangonels are onagers.

Onagers and mangonels differ from trebuchets in that trebuchets use counterwieghts, have a longer arm and are intended to hurdle things over walls.

Onagers and mangonels are the same in that they are smaller, more compact and are intended to hurdle projectiles through walls. They have a lower trajectory and higher velocity than a trebuchet.

An onager uses a sling to hold the projectile, and relies on torque from a twisted rope to hurdle the projectile.

A mangonel uses a fixed bowl shaped bucket on the arm to hold the projectile and a counterweight with pulleys or gears. The mangonel could fire off more rocks than an onager.

Wikipedia has some pretty good crosslinks and image of the different types of catapults. and if you create a grid and compare the different types, you will be able to see how the three are the same/different. I think it’s easier to understand with the trebuchet in the mix. Compare the mangonel to the trebuchet, then compare the onager to the trebuchet. These comparisons have similarities and differences from each other.

The history of the words is something like the following:

Manganon – ancient Greek word for war engine
Manganum – latin form of Greek root word
Manganeau – Pre-medieval French form of word
Mangonel – Medieval English form of word
Gonnel – late medieval form, as gunpowder weapons were being invented
Gun – post-medieval form

Greek onas (ass, the animal) + agros (field) = onagros, meaning a wild ass (animal)
Onagrus – Latin form of Greek onagros, now referring to Roman design which had a scoop and cord torsion for power.
Onager – English form which refers to the Roman form of catapult they called Onagrus, or medieval versions similar but somewhat different to the Roman design.

Onager is a little more specific that Mangonel, though there are variations on the Onager. Mangonel means war engine, and may be used by some people to mean certain types, which may be different than other people’s meanings, especially if you compare French versus English.