The first Stewart King of Scots was Robert II, grandson of Robert I, The Bruce, and son of Marjorie, the Bruce's daughter, and Walter the Steward. He inherited the crown because his uncle/ Marjorie's brother/ Robert the Bruce's son, David II, had died without issue. So hardly stolen.Don't trust those High Stewards, they have been known to steal the throne and then change their name to Stewart and then to Stuart! A smarmy lot!
James VI and I inherited the throne of Scotland from his mother Mary, Queen of Scots, and the throne of England from both her and his father Henry Stuart, Lord Darnley, both of whom were grandchildren of Henry VII through his daughter Margaret Tudor (Henry VIII's older sister). The spelling Stuart seems to have been adopted by Darnley to distinguish his branch of the family from his wife's, and it was inherited by his son, who probably preferred it for similar reasons, it was politically expedient to re-fashion himself as a new-style monarch of a newly United Kingdom, distanced from the Stewarts in Scotland and the Tudors in England.