at home is just a special idiom. If you want to emphasize staying inside, that is, the interiority of the situation, use
in the house.
If you want to emphasize that you don't want to leave the house (Maybe
there's a dangerous animal just outside the door!), then say that you
want
to stay in the house. If you want to emphasize that
you don't want to participate in social activities in the world at
large (shopping, going to a restaurant, going to work, etc.), then say
you want
to stay at home.
You can say
at the airport when you are in the airport.
You can say that your dog plays
at the playground.
home,
airport, and
playground can be conceived of as places where certain typical activities occur. When they are conceived of in this way, we use
at.
When we conceive of such places simply as delimited volumes -- i.e., objects with interiors (
house, airport) -- or delimited areas (
field), we use
in.
CJ