A double jump definition is essentially 'jump, then once in the air you can jump again with no extra skill'.
The first I saw that had it was Super Ghouls & Ghosts, but that's probably not the first that had it.
You jump, then you can jump again as if you were in an air platform.
Space Jump (Metroid II: Return of Samus) sort of counts as multiple/infinite jump... but the first jump must be a spin jump, and it also requires timing, so it's not quite the same.
The Cape (Super Mario World) wouldn't count because you need a running start. It achieves a similar effect to Space Jump but this time with pressing left and right to pivot Mario/Luigi's flightpath. Staying aloft requires some skill, and actually rising requires even more. Plus you need a big open stage for it.
Flutter Jumps count as an extended jump, I suppose (Yoshi's Flutter Jump, Luigi's Flutter Jump in Super Luigi U), whereupon holding the jump button has the character perform some kind of effort in order to stay in the air.
The last type I would imagine would be an air dash. Games with Booster Jumps/RocketPacks/JetPacks (Cave Story) follow this dynamic. MegaManX1 and X2 gave you the ability to dash in the air by pressing a different button altogether, and MegaManX3 gave you the ability to use the dash plus a direction, essentially giving you a complicated double jump by pressing UP + Dash.
These days it seems every game wants to give you a doublejump, and while that may be acceptable, not having an explanation for it is really immersion-breaking (I believe Leon Belmont can do it by default... or was that Hector? I don't remember). It's starting to feel lazy and a little grating.