From how I was taught, the spade refers to the actual spade mouthpiece(spoon & braces) regardless of the cheek style. The spade refers to the spade because it is not considered as or to be ridden as a typical leverage style bit, but strickly as a signal bit. Other mouthpeices like the Mona Lisa(which I have not found many horses to like), the Frog, San Joaquin, Half Breed and such can all be ridden like any conventional leverage bit. The Santa Barbara style which you descibed above is a typical cheek style because of the balance it provides for the mouthpiece, not the mouthpiece itself.
I DON'T LEAD 'EM AND FEED 'EM, I RIDE 'EM AND SLIDE 'EM.