The first definition is wrong, A cappella is not using any instruments but the background music is usually accompanied by beats made by the human mouth, usually one linking to the other one, like hitting the floor, then the desk, then the wall to make a beat that goes together, then it is accompanied by the actually singing, A Capella can only be accomplished by separate recording of different sounds, or a group working together
Go to youtube and check out The Temptations-My Girl-A Capella(MrVoxter)
Check this out, this is an A Capella and a really good one
A genre of music without there being any actual background music. It is usually used by rappers.
At the end of the last rap battle in 8 Mile, B. Rabbit says: "**** the beat, i go a capella" and carries on rapping without the backing music.