The battle between "Myanmar" and "Burma" is a very interesting naming geography battle. First, the naming battle is only in foreign languages and not in Burmese. Both the pro-Myanmar and pro-Burma name camps recognize the formal written form of the country as Myanma (Myanmar) while Bama (Burma) can be used as the primary spoken word choice for the country's name.
Both words have their origin in referring to the ethnic Burmese people. "Burmese" in this case referring to the actual ethnic group and not all people found in Burma/Myanmar such as Shan, Karen, etc. Burmese, like many Southeast Asian languages, has different words for the same noun in written and spoken use.
After independence in 1948, various governments have tried, sometimes/mostly half-hardheartedly and intermittently, to use the world "Myanma" to refer to all people in the country while keeping "Bama" to refer specifically to the ethnic Burmese people.
In 1989, the military dictatorship conducted a review of English place names in order to rid the map of colonial influenced spellings. One of the decisions the naming committee made was to change the English name of the country from "Burma" to "Myanmar". The reasoning was two-fold 1) Burma referenced the spoken and not written form of the country which was viewed as informal and 2) the committee stated "Myanmar" was more inclusive and showed that the country had many ethnic groups comprising one nation while Burma implied a nation-state to the exclusive of a third of the country.
Many opposed the official English-language name change of the country. Most international opponents claimed the military junta was illegitimate and therefore the name change was void. In country opponents meanwhile stated that minorities recognized themselves as part of a Burmese nation since they spoke the "Burmese" language.
The truth is a bit more complex. A survey of minority resources seems to indicate that most but not all tend to actually prefer the term "Myanmar" though the usage of Myanmar to English-language audience is complex because some believing using "Myanmar" implies support for the military/post-military government when it actually does not necessarily imply support.
While the military/post-military government puts on a good face, it is hardly minority friendly. The government engages in a sort of militant national (Burmese) Buddhism which views minorities, especially Christians and Muslims as potential threats. Government-minority issues are further complicated by the fact that the country has undergone 63 years (since 1949) of conflict in which every in-power government has fought with and against every significant minority group in the country.
Meanwhile, the anti-military/post-military, pro-democracy ethnic Burmese forces are not so minority friendly themselves. The South Asia blog Bangla Nation pointed out that Nobel Peace Prize winner Aung San Suu Kyi is a Burmese nationalist at the expense of minorities.
So is "Burma" and "Myanmar" more inclusive? Take your pick. In reality the naming issue is more of a distraction. The complete failure of tolerance and ethnic acceptance in the country is the bigger problem. However, geopolitics right now dictates that the world powers will focus on the post-military government's relationship with the People Republic of China rather than internal issues like a bloody but outside the press' interest civil war.