Either preposition could work in all of your phrases; it depends on the context and the perception of the speaker and the location of the facility. At refers to a point reference ( He's not home; he's at the shop ) and in refers to a 3-dimensional situation ( He's not in the shop; he's in the storeroom ).
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Chariotif the above phrases do not contain "the", there is less sense on the physical side. Is this perception correct?Yes, that is the general idea.
Mister Micawber.What did you mean by [at least re at the place in AmE]?
Not quite accurate, Sitifan, at least re at the place in AmE-- both are at and in are fine. See CJ's comment on context.
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the meaning 'at the place, not at home' is expressed by [1] or [2] in BrE, and by [1] in AmE.