I think it might be useful to differentiate these three terms. Marxism was an economic theory. I saw WAS, since Marx offered a scientifically formated prediction–a hypothesis–which was falsified by history. He was wrong.
Specifically, he predicted that wealth, being finite, would continue to concentrate in the hands of a few, impoverishing thereby the many, who of course could not be expected to put up with it forever. Revolution, according to this hypothesis, would NECESSARILY occur in the nations where this wide gap first emerged, namely the already industrialized nations.
Yet, in Russia, 85% of the population lived on farms. Only perhaps 10% of the population, on the high side, was even remotely in the “proletariat”. Everyone else fed themselves and their families–and feudal lords–with no interference from anyone. Lenin tricked them into supporting him by promising them land. Mao did the same thing, while contradicting Marxist ideas even more dramatically by STARTING in the countryside. He was helped greatly by leading an effective anti-Japanese guerilla war, and, again, by promising the peasants the sun, moon, and stars.
What needs, therefore, to be added to this mix is Lenin’s notion of the “professional” revolutionary. Marxist doctrine held that the revolution was inevitable, and could not be hastened or forced. Lenin taught that a small cadre of people dedicated to cynical deceptiveness, ruthlessness, and above all a PLAN, could take power, in the NAME–not the reality–of Marxism. Hence the frequent use of Marxist-Leninism, which is an oxymoron.
Leninism=Communism, which is a POLITICAL form, not an economic one. It is one which uses POWER backed by and often signified by intentional terror to cow the masses into silent compliance.
Socialism is a CULTURAL form. It is a moral claim that inequality is wrong, and that the MEANS of centralized power–there is a continuum here, but the intent remains the same–is how you fix it. It deduces from the Mercantilist fantasy that wealth is limited, that wealth concentrated is necessarily wealth stolen, from which it rationalizes the theft of property by the State. It provides the rhetorical cover for totalitarianism, which is why socialists are utterly unable to condemn the abuses of Cuba, China, and North Korea. As long as the only crime is inequality, then anything that addresses it is moral.