I agree that the Wade Giles of pronunciation is confusing, but it is not responsible for the different pronounciations.
Peking (pronounce bayjing) means 'northern capital' and is used when Peking is, in fact, the capital. In times of history when China had a different city as its capital the name of the city was changed to mean to mean 'northern peace' and is pronounaced bayping (bay meaning northern and ping meaning peace).