1. The Sepoy Mutiny : This name is rarely used in India itself. Strictly speaking it is only part of what was happening in India at the time. The rebellion is popularly believed to have been sparked by the mutiny of Indian soldiers in Barrackpore. Mangal Pandey, the soldier who started it, is considered a freedom fighter by Indians. The mutiny spread quickly and several other units also took up arms against the British. They even gained a lot of initial success and took several important cities like Delhi and Lukhnow. This description however ignores/overlooks the contributions of the other people involved in the fighting against the british. The soldiers were only a part of the forces against the british. Several of the petty kings had grievances against the british and took up arms, as did several of the zamindars who had been evicted by british actions. The biggest component of the forces however were the poor peasants who had suffered greatly under british rule.

2. The First Indian War of Independence : This is the name given to the events when they are taught in Indian schools. It is therefore a popular name for the events in India. It is again not entirely inaccurate. The uprising was popular among the indians. The fighters came from all walks of life going beyond traditional divisions of caste/religion/language. It really was a war by the people to rid themselves of the british. The level of national unity being displayed was unprecedented in the country. This name however tries to encompass too much. The uprising was not as popular as Indian schools books lead us to believe. Large parts of the country did not participate in it. Delhi was recaptured by a column of indian troops from punjab. Other strongholds of the rebels were again retaken by the british by using Indian troops. Several of the rulers sided with the British instead of the rebels. This siding was part of the reason the British abandoned their doctrine of lapse to annex the kingdoms. many of the zamindars who had sided with the rebels initially switched sides later when the britsh agreed to grant them some of the lands that had been taken from them. Most importantly though, the rebels did not have an idea of India that they were fighting towards. The fighting was aimed at getting rid of the british and reverting to the old systems. That was the reason they rallied around Bahadur Shah Zafar, the last mughal emperor. This was also the reason that the rebels did not fight as a united force. Once they succeeded in getting rid of the british in their local area, they would not go to help the rebels elsewhere. Why would they, they were not thinking of the rebellion as a war to unite india, they were just getting rid of the british opression on themselves.

3. The Indian rebellion of 1857 : I find this to be the most accurate of the three that you have mentioned. It portrays the events just as they were. It was a rebellion against the british not a war for independence by the indian people. At the same time it does not reduce the events to a simple mutiny by some of the soldiers, ignoring the contributions of the rest of the rebels.

Source : India's Struggle for Independence by Bipin Chandra