A Marxist follows, by definition "a theory
or system of social reform which contemplates a complete reconstruction
, with a more just and equitable distribution
, Webster 1913
). Therefore, a Marxist
is always a socialist
While it is also true that a Marxist is generally a follower of the beliefs of Karl Marx, that is also not entirely true. Marxists generally take much from the works of Marx, but they very rarely follow, word for word, the dogma of Marx (which, since he was primarily a philosoper, though an active philosopher, was fairly theoretical and impractical).
Marxist is also a title that is given to the followers of one particular strain of socialism (which, incidentally, existed long before Karl Marx) that emphasizes violent revolution as the only way of brining about a truly equitable and fair future, a future that they call Communism.
New socialists, by the definitions used by both Marx and Lenin (who, some say, simultaneously held true to Marx's work AND furthered it), are clearly NOT Marxist. New socialists are revisionists - people who believe that the proletariat revolution can be done peacefully, by taking over the government through legal, peaceful means ... primarily, the ballot box. Marx, in 90% of his writings, emphasized the need for real revolution - that the revolution had to happen through fighting, not through voting.