Fiat Lux!
The classification originates from those smarty pants at Harvard:
http://en.wikipedia.org/wiki/Stellar_classification
It's been extended over the years and, like many classification schemes in astronomy (in my experience), it's a living thing as more objects are discovered that don't quite fit.
http://en.wikipedia.org/wiki/Brown_dwarf
Covers your very question - it suggests (although no reference given) that an object must have sustained fusion at some point to be counted as a star - seems reasonable.
Biting the hand that feeds IT
