Movement of National Liberation was created in 1961.
Arab Liberation Movement was created in 1952-08.
Antigua Caribbean Liberation Movement was created in 1968.
National Liberation Movement - Albania - was created in 1942.
Movement for the Liberation of the Congo was created on 2003-04-05.
Democratic Movement for National Liberation was created in 1947.
Croatian Liberation Movement was created on 1956-06-08.
Popular Movement for National Liberation was created in 1972.
National Movement for the Liberation of Azawad was created in 2011.
National Movement for the Liberation of Kosovo was created in 2004.
Revolutionary Liberation Movement Tupaq Katari was created in 1985.
Yes, they mean the same thing. Animal rights is the ideology behind the animal liberation movement. Animal liberation is the ultimate goal of the movement, and the latter term is used more often because I think animal liberation sounds more radical and unsettling.