A musician is also someone who writes music, even if they write it for other people to play. People who write music are called composers. Usually that part is left out if they also play or sing the music they write, but they are still composers because they wrote the music.
Musicians can also make a group together to play songs. If the group has many people playing instruments together, like Beethoven's music, it is called an orchestra. If it is just many people singing, like in a church, it is called a choir. If it is only a few people together it is usually just called a band. Sometimes a band has the same name as the singer.