According to etymonline, Yankee has been used to refer to different sets of Americans by different people for hundreds of years.
1683, a name applied disparagingly by Dutch settlers in New Amsterdam (New York) to English colonists in neighboring Connecticut. … In English a term of contempt (1750s) before its use as a general term for “native of New England” (1765); during the American Revolution it became a disparaging British word for all American natives or inhabitants. Contrasted with southerner by 1828. Shortened form Yank in reference to “an American” first recorded 1778.
The British calling someone from Texas a Yankee isn’t really any more right or wrong than someone from Texas calling someone from Pennsylvania a Yankee. Words can have contextual meanings.