I believe that at the core this is prescriptivism as applied to orthography, though more examples would be needed to derive a definitive conclusion. In your first three examples1, these are official in some way or another: Kolkata and Mumbai were officially changed by India's government to better match local languages, while Beijing is the Hanyu Pinyin romanization, the official romanization scheme according to the Chinese government, and also later by ISO.
The case is similar for Cambodia. In 1989, it's name was officially changed to the State of Cambodia from the previous People's Republic of Kampuchea. This change was made while the Vietnamese-backed government was still in power. Its unclear, but based on my brief readings, it seems the change was part of wider reforms which were aimed at restoring Cambodia to pre-Khmer Rouge times and improving international image. In 1993, the monarchy was restored, and along with it the official name previously used under the king, the Kingdom of Cambodia.
English does not automatically have to accept official declarations from countries speaking other languages (note that English is an official language of India however). However, unlike other languages, such as French and German, English has no comprehensive language academy such as the Académie française or Rat für deutsche Rechtschreibung. This might give English a better chance at accepting the official declarations of countries speaking other languages. Compare:
- English: Beijing, French: Pékin2, German: Peking
- English: Kolkata, French: Calcutta, German: Kalkutta
- English: Mumbai, French: Bombay2, German: Mumbai
Of course, there will always be exceptions. For example Ivory Coast requested they be referred to as Côte d'Ivoire with little success outside official diplomatic communication. Additionally, there's the Burma → Myanmar change, with partial acceptance since that government's legitimacy is questioned. As for Cambodia, it only ever used "Kampuchea" officially between 1975 and 1989.
- Iraq's a little different in the sense that there's no orthographic change here.
- These are officially recommended.