Why did Lady Jane Grey end up headless?

Her 'reign' didn't make any sense, because the natural order of succession was reinstated after those pointless 9 days anyway. Was she just an innocent pawn used in the royal politics of the day, or did she really feel that she was rightful heir to the throne? She's often referred to as 'Lady Jane Grey', not very often 'Queen Jane' - was that because she wasn't recognised as a 'real' queen, .....was she seen as a usurper? At best....it's 'Lady Jane Grey, the 9 day queen'. To my mind, a queen is a queen - whether it be for 9 days or 9 years.