"Bangladesh War" is a common name for this conflict, but this term is also used for the eastern front of the Indo-Pakistani War of 1971 war, and is generally understood to be coterminous with The Indo-Pakistani War of 1971 (see below).
"Bangladesh War of Independence" is the most commonly used name outside of the Indian subcontinent. It is a common name formally used to describe many other successful secessionist wars (see list of War of Independence).
"Bangladesh Liberation War" (Mukti Judhho in Bangla) is officially used in Bangladesh by all sources and by Indian official sources. The proponents claim that having won 167 out of 169 seats of East Pakistan, the Awami League had a popular mandate to form a democratic government, and this gave Sheikh Mujibur Rahman, as the leader of the party, the right to declare independence of the country. In Bangladeshi eyes, since Major Ziaur Rahman claimed independence on behalf of Sheikh Mujibur Rahman, an independent Bangladeshi government was in existence as early as 26 March 1971, and therefore the war was fought by this government for the liberation of its territory.
This nomenclature is politically preferred by both India and Bangladesh for a few reasons:
"Pakistani Civil War" describes either the period of March 26, 1971 to December 16, 1971 or the period of March 26, 1971 to December 3, 1971. However, it is rejected by Bangladeshis who dislike the association with an internal struggle of the state of Pakistan.
"Indo-Pakistani War of 1971" is most commonly used to describe the period between December 3, 1971 and December 16, 1971. The Indian Army does not explicitly use the term to describe the war in their Eastern Front at any point. Instead, India only refers to the war on the Western Front as the Indo-Pakistani War. (Note that the Indian Parliament recognized the People's Republic of Bangladesh as an independent country on the 6 December 1971.)
The proponents of this terminology also question validity of the declaration of Bangladeshi independence since there was no foreign government that acknowledged the independence; thus the war was effectively between Indian Army and Pakistan Army.