A summer house, a cottage or a cabin?

The generic term in British English is holiday home (this is more common than summer home) - this covers all types of building. Cottage and cabin are types of building. A camp is a place/area where people stay, rather than a building or home.

I'm not clear whether you're suggesting that summer home/cottage suggests small buildings in gardens, or whether you're referring to cabin & camp in that respect. We certainly wouldn't use the latter two terms (unless cabin were an accurate description of the type of building).

[Edit in response to OP's comment: Summer house might mean a small building in a garden (something which I overlooked and which, I think, would probably be more commonly used in the context of a large house with a large garden), but summer home wouldn't mean that.]

I think we would often just refer to it as a second home or even something like They've also got a place in [village/town/county/country name]. Both terms could also apply to places abroad, particularly places in more southern parts of Europe, which may be used more in our winter than in our summer.