In rough order of altitude expected...
Human tow. Get some young strong folk, use a speed changing pully/block arrangement, and tow to the limit of the field and crew endurance. Fairly short altitudes, but a marginal powered craft can cruise away once airborne.
Bungee tow, restrain craft and have humans run with multiple tow ropes with elastic sections. Release craft before you drag the crew backwards in a cartoonish way. Very typical system for hill launch popular in 1920-30s.
Tow by fixed location pull winch. These run from mini trailer mounted scooter tow, for hang gliders and paragliders, to big sailplane rigs.
Truck/car/boat tow with fixed length lines. I rate this as most dangerous, ymmv.
Truck/car/boat tow with payout winch. The safest and best way, IF you have the ground room.
I've used every one of these, you may argue that "catapult" doesn't apply to truck/car /boat tow, it's a question of definition.
things I have not personally tried, but know exist.
Explosive launch catapult. Most famously used in CAM ships to launch Hurricane fighters one way to defend the convoy. Explosive charge piston linear.
Rocket launch catapult. Most famous in the second generation of CAM ships, and Zero Length launch systems for jets in the Cold War.
Steam catapults. Aircraft carriers from pre WW2 to today.
Linear motor electric. Latest Carrier system.
You missed gravity launch down a short runway on a mountain top.
Horse tows, too -