this term seems to have morphed from only having a negative connotation to general use. It is difficult (impossible?) to find older uses of this word that don't imply something negative.
► uses
Uses:
Thou makest us a byword among the heathen.
Ps. xliv. 14
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This, in truth, his neighbours might have pardoned, seeing that saints have never flourished in those parts, but there was in him a certain wanton and cruel humour which made his name a byword through the West.
Arthur Conan Doyle. Sherlock Holmes: The Hound of the Baskervilles (1902)
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"I know you and Harry are inseparable. Surely for that reason, if for none other, you should not have made his sister's name a by-word."
Oscar Wilde. The Picture of Dorian Gray (1890)
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Michael Portillo: George Pullman built this factory in 1880 to manufacture carriages for his Pullman Palace Car Company. In Appleton's day they became a by-word for luxury travel.
BBC. Great American Railroad Journeys: Chicago to Champaign (2017)
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A clever, wide-awake business concern should be able to make three-quarter-length trousers a byword of masculine fashion.
John Kennedy Toole. A Confederacy of Dunces (1980)