Some of it would depend on context (though I'm guessing your context is Girl Scouts
Sleepover - At someone's home. I would expect to sleep in the bedroom or living room, either in beds, on sofas, or on the floor. All of the comforts of home, minus perhaps sleeping in your own bed.
Overnight - I'd associate this more with a youth group or something. Depending on the setting, they might be in a church sanctuary, gym, or in a dorm-type setting. I would expect a bathroom, perhaps some access to a kitchen, and some reasonable climate control and shelter from the elements.
Backyard Camp - Pitch a tent in someone's back yard. Unless otherwise specified, I'd expect one to two tents (i.e. a small group). Access to a home kitchen and bathroom and able to come inside if the weather gets nasty, a kid gets scared, etc.
Cabin Camp - an organized campground (e.g. a 4-H camp). Cabins should have bathrooms (or perhaps a more central "bath cabin") but I'd expect flush toilets and beds/bunks of some sort. I would not expect linens to be provided. Cabins may or may not be climate controlled, but I wouldn't expect to be in an unheated cabin if it's bitterly cold. If it's a summer camp type setup, I'd expect a dining hall. If it's more of a national park setting, I'd expect some access to kitchen facilities or an indication that only grills are available. A more primitive cabin camp, especially at a national park, might have long drop toilets. I would not expect a long walk to reach the cabin unless specifically advised otherwise.
Camp - a catchall term, but I'd probably expect a tent.
The tent might be pitched in someone's back yard, at a campsite where you park next to/near your tent, or it could be in the back country somewhere either on- or off-trail. I would not expect to sleep in a permanent structure, unless advised otherwise. Expectation of access to electricity and running water would be dependent on the context.
Primitive Camp - I'd probably just call it camping.
I would expect to carry everything I need, find a site suitable for camping, and pitch a tent. I would expect to use portable stove for cooking and would expect to find and purify water. I would not expect electricity or a flush toilet.
I would not necessarily expect easy access to a road.