See the link below for more information on the dance The Spanish Gypsy (or Jeepsie), the song from which the tune for the dance came, and the 1623 play from which the song came, which had the title "The Spanish Gypsy".
I'll go out on a limb and make some historical pronouncements which cannot be proven, but which seem most probable to me:
The dance title The Spanish Gypsy came from the dance being done to a tune associated with the play The Spanish Gypsy.
The dance figure Gypsy got its name from the prevalence of the figure in the dance The Spanish Gypsy.
The Morris dance figures whole-gyp and half-gyp were originally called whole-gypsy and half-gypsy. (Although parts of England had and ancient tradition of seasonal dancing under the name Morris Dance, it seems likely, from the nature of the dances, that the form of the Cotswold dance traditions collected by Cecil Sharp only went back to the Elizabethan period.)
I offer the above hypotheses to counter the claim that the dance term "gypsy" was based on an ethnic stereotype. Of course, even if I'm right about these hypotheses, they have nothing to do with the fact that the term "gypsy" offends some people, which we want to avoid.
Jacob