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The term multiple star system (or multiple star) clearly means at least two stars co-orbiting. The term is more often used if there are more than two, the shorter, more common term binary star used for pairs of stars. The term higher-order multiple star system is sometimes used to explicitly indicate a system of three or more stars.
Cases where someone might deliberately use the term binary star when the system has three (or more) include the case where the system was known to have two stars for substantial time before an additional star was discovered. The term binary star might also be used in the case where a pair of the stars have some particular interaction that is of interest, often when a pair of the stars are close together and a third is widely separated, such as Alpha Centauri. Such an interest might concern the close pair, e.g., regarding mass transfer, but in some cases the pair of interest might include the more distant member, e.g., considering their large orbit and proper motion in cases where the other two are extremely close, e.g., a contact binary.