As far as I can tell, the name is dependent on the garnish. Effectively a green French Martini and this drink is definitely one for those with a sweet tooth. As with the French Martini the quality of the foam is mainly dependant on the quality of the pineapple juice.
I served this for a Halloween dinner party with a crucifix on the side, stuck into the cocktail cherry and with a drop of grenadine to add a red spot on the foam. (The grenadine then sinks into the bottom of the martini glass giving a red base.)