Do these toys have a common widely-accepted name?
Inflatable - noun
1 - any of various large air-filled objects made of strong plastic or rubber, used for children to play on at fairs, carnivals, etc.
Here are a few usage examples from around the US:
- New York Post: Bounce house goes airborne with kids inside
- NBC Miami: Children Hurt as Waterspout Uproots Bounce House
- Denver Post: 2 kids injured as Colorado bounce house blows away
Variations like bouncy house and inflatable bounce house are also used, but less commonly than bounce house.
Every child I grew up with would refer to them as a moon bounce.
They have been marketed with such names as "Bounce House", "Bouncies","Moon Bounce", "Boingalow", "Astrojump", "Moonwalk", "Jolly Jump" and "Spacewalk". "Brinca brinca" [is] another name commonly used by Latinos
Worked for about five years at accompany that rented these out in the USA, and they were called jumping castles (when they resembled castles, and were for jumping in) and inflatables the rest of the time. This includes slides, Velcro walls, or obstacle courses as your picture indicates. The company has them listed generally as 'inflatables' on their site.
Note that jumping castles or bouncy houses are simply a more specific for of inflatable. The following photos are all from their website.
Jumping Castle, Bouncy House, or simply Bounce
45 foot obstacle course Inflatable
27 foot dual-lane Inflatable slide
I would add that there may also be regional differences. My experience was all in the southwestern USA.