As a modern subset of the generic "Climbing Frame" (this being AmeriPol, I don't begrudge the term "Jungle Gym" - though I have never heard it used outwith the all-ervasive US media/reference... I don't think it has seeped into UK culture) I've never heard of a specific name for the dome types.
(Back in my day, climbing-frames tended to be almost always the exact same purely cubic framework, maybe of varying size/complexity but (IIRC) four or five 'cubes' high in general, five or size 'cubes' square in plan with a further inner fifth/sixth level in the middle of the top. All set into the ground in a huge slab of rough concrete, sprinkled loosely with broken beer-bottle glass. Greater imagination didn't tend to happen than that. Except for what colour to paint the welded-solid scrap steamroller they might also have stationed near to the swings and slide.)
So, "dome-type {climbing frame|jungle gym}" would work with me.
"Monkey bars", for me, is the (sub-component?) frame that's the elevated 'horizontal ladder' that you're supposed to swing-walk, dangling from your arms, often as part of a more imaginative 'course' (connected; or disconnected but in line, with on/off-zones between that and the balance-log, climbing wall, pillar-tops, etc) or as an optional limb/mid-connector of a multi-activity climbing-frame assembly.
But I imagine that there's all kinds of names used, given kids' immediate need to identify such things among their peers (with greater or lesser adult cultural influence), away from any immediate contact with the all-knowledgeable adults, or the official play-frame sales catalogues.