The American circus does have a different back-story for the origins of the word "bulgine." (fwiw it's not in my two editions of Webster's. Go figure.)

The UK quote above is lifted from an old minstrelsy comedic routine known as a Locomotive Lecture. It was generally performed as a companion piece to the Locomotive dance that imitated the new steam engines.

See also Lighter's old post here:
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Dancer George "Great Western" was one of the first to develop the form. He was also the father of the Star Sisters, Lucille & Helen Western, who followed into minstrelsy after his death. John Wilkes Booth supposedly carried Helen Western's cabinet card in his pocket.

William M. "Billy" Whitlock toured with the "Great Western" in 1843 as a member of the Kentucky Minstrels (banjo & comic lecture.) By 1845 Whitlock added his own dancing for the Virginia Minstrels tour. It was one of his regular routines for several years.

Both ensembles spent part of their 1843-45 seasons in London so 'bulgine' ("Great Western" &c) may be a play to British audiences but the 'real' etymology would have been more like Yogi Berra meets Abbott & Costello.