Here's a helpful definition and description of what a Composer really is.
A composer is a person who writes music. The term refers particularly to someone who writes music in some type of musical notation, thus allowing others to perform the music. This distinguishes the composer from a musician who improvises. However, a person may be called a composer without creating music in documentary form, since not all musical genres rely on written notation. In this context, the composer is the originator of the music, and usually its first performer. Later performers then repeat the musical composition they have heard.Â (From Wikipedia)
a person who writes music - notice that the "writing" music using pen and paper is not always necessary, as indicated later in the paragraph.
particularly to someone who writes music in some type of musical notationÂ - this points out that in the old days, composers did "write" music in some form of common or recognizable musical notation (using pen and paper, too)
notice that writing the music allows "others to perform the music" - this is key in distinguishing the composer from the musician; nevertheless, musicians can also be composers, if they are...
the originator of the music, and usually its first performer - aaaah...the originator...this is key, especially for our day and time in which so much music is "composed" using computer hardware and software; don't be surprised to find out that many composers don't know musical notation at all!!
How is a music composer different from a music producer?
They are very different in many ways. However, it is common to find composers who are also producers and vice versa. The roles are very different, though. While a composer focuses specifically on creating a musical piece, a producer has a much broader focus related to the creation and direction of a complete musical product like an album/cd, for example.
How is a music composer different from a beat maker?
While there are differences between a music composer and a beat maker, keep in mind that the ROLE of a music composer and the ROLE of a beat maker are pretty much the same. Both the composer and beatmaker focus specifically on creating a musical piece.