Um, Peking comes from the Cantonese pronounciation of the name. Beijing is the Mandarin pronounciation. The reason the Cantonese version was used originally is (1) China was originally governed from the south and (2) because the majority of the Chinese who emigrated and settled around the world at the time (1800s to early 1900s) were from the south and their native dialect was Cantonese.

Given that Mandarin is now the national language, it makes sense to use the Mandarin pronounciation. I'm not sure how this was interpreted as some sort of political correctness action.