Captivity: normally referring to animals, such as animals in a zoo. I would never use this when referring to people actually. Detention: Refers to being bound in one place, could be an official sentence, "home detention", or even kids in school detention. Custody: Being in 'custody' refers to being held/imprisoned by someone. You are "in their custody".

To illustrate it a bit more, in sentence 1 the prisoner sounds like a victim. In sentence 2 and 3 it sounds more like they were arrested for breaking the law.

Same general meaning but used in different situations. Captivity is usually used for unlawful detainment like slavery, kidnapping, or even POWs. Detainment doesnt always mean imprisonment, although it can. You would usually use that word for something more temporary like being held at an airport, or even someone making you late to something. Custody is what you would use for someone under arrest or in prison.