There's an interesting story behind No. Eleven, only sketched in the Museum's description. The original color of the flag was black (now faded to brown, except for its battle honor, WINCHESTER). It had been captured or "acquired" by Confeds, and was presumably lying about a camp site when re-captured (or acquired) by Detective March, for which he originally rcvd the Medal of Honor, later recalled. It was displayed after re-capture as a Rebel "No Quarter" flag from one of the several battles at Winchester. Only after the war did a former Union signal man come forward and explain that, no, this wasn't a rebel flag, but a highly prized Union Signal Corps "Battle Flag," awarded by the Chief of the Sig Corps to a Signal Officer for duty performed under fire, the name of the engagement being printed on the point of a star. (Unfortunately, the number of battles came to exceed the points of the star design, and that aspect--adding the name--was discontinued early in 1863. Note that this was a personal award, not a unit one, perhaps the equivalent of a Silver or Bronze Star today. As for an NCO or EM, the best he was offered was to have his name reported to the General-in Chief, and maybe a promotion.)
Evidently the explanation of these relatively rare "utility" flags never quite "took" in officialdom. And when the "rebel" flags were eventually returned to their home states (Virginia being the default), this signal Battle Flag was included. A little research might identify the Union officer originally awarded this flag (but perhaps it would be an embarassment, for it would also show who presumably "lost" it later).
In the modern US Army's regimental heritage system, the "Signal Corps Regiment" [sic] displays the star flag in its insignia.