in / at "home" always applies to one's own house / living quarters / living area / place of residence. Example: "he's at home" (he's at his house / living area) "I'm at home" (I'm at my house / living area) "I'm home" (same as above but different grammar) in / at "home" is the most informal in / at "the home" is less informal in / at "the house" (or "apartment", "living quarters", "place of residence" etc...) is the most formal. I personally don't hear in / at "the home" very often.. usually only when we are being informal about somebody we don't personaly know. Ex) "The police are at the home of the suspect" I would say that's probably the only situation where we use "the home", it might vary regionally. It is however the rarest, because the word "home" is very informal by its nature, for formal we usually just say (house / apartment / place of residence etc.)

Mainly I would use "the home" to talk about a home that is an institution, like "She lives at the home for troubled children." Another use can be like previously stated, with the police or a legal situation. Saying "I was at the home of a friend" sounds a bit unnatural or formal and I would be more likely to say "I was at a friend's house." Usually "I'm at home" "He was at home" is more natural and common.