Marxist, in general refers to someone who studies, learns, analyses, accepts and acts according to a wide socio-economico-politico-philosophical range of ideas & theories that were introduced by Karl Marx and Friedrich Engels, mainly on their analysis of the capitalistic system and their introduction of scientific socialism & their analysis towards the communist society.
As you mentioned, there are several Marxist currents, due to the fact that Marx's analysis & thoughts has affected many economists, politicians, analysts, etc. Thus, there were certain groups that accepted parts of it, or further analysed it, or went to different directions or even denied it, leading to different schools.
The most influential one was of course Lenin's approach, who is also considered the orthodox one. Lenin was a Marxist and as a Marxist, he further analysed how the capitalistic state —controlled by the ruling class— oppresses the working class, how ruling classes of other countries oppress the working classes of different countries through imperialism and also gave more emphasis to the practical aspect as to how to battle capitalism and to build a socialist society, which will in turn lead to a communist society in the future. His ideas and actions were also the inspiration for the Russian revolution and afterwards towards the building of the socialist Russia -->USSR. In a simple and quick description, this is mainly referred as Marxism-Leninism or orthodox-Marxism, or "the practical view of Marx's ideas".
A Marxist-non-Leninist, will be in one hand close to Marx's ideas, but denies/argues with/has a different view than Lenin's approach. Examples of this can be Anarcho-communists or certain Trotskyist groups.
Usually, the main point of conflict is the way true communism can be achieved. While Lenin definitely puts the socialist state as a prerequisite and sets a great number of analysis on the subject, there are Marxist groups who totally deny it and aim towards a free-communist society as an end-target.