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The term Be star indicates a B-type star with emission lines (as per the spectral type suffix "e"), but is commonly used to indicate those with a particular frequent underlying cause: hydrogen emission lines presumed to stem from rapid rotation, producing a circumstellar disk of gas ejected by the rotation (a decretion disk), and heating it to produce Balmer series emission lines. They often show overlaid H-alpha emission and absorption lines, each with considerable line broadening, the former indicating the rotation of the disk and the latter, the rotation of the star. The term classical Be star is used to distinguish this from those with other characteristics, presumed to be formed by some other mechanism. The term Be star is now so commonly associated with the rapidly rotating type that such rotating A-type stars and O-type stars are sometimes referred to as Be stars.
The term B[e] star specifically indicates forbidden emission lines, suggesting some difference in the underlying cause or some additional factor.
Note that Herbig AeBe stars have the same spectral classes and have Balmer emission lines, but are a type of pre-main-sequence star (PMS) and I believe their characteristics do not result from rapid rotation.