Your definitions of Socialism and Communism are almost entirely false and lacking in any reference to the founders of the movement, their usage in the disciplines of philosophy, political philosophy and sociology, and historical meaning.
Socialism since its inception was never about redistribution of capital gains, it was about the ownership of society's means of production by the working class, i.e those who survive mostly or entirely from the sale of their labour-power. This definition stretches back to even before Marx, as does Communism. Your definition of "Socialism" is what it is thought of in the US, i.e Socialism is where the government spends money on social programs which it acquires via heavy taxation. This, as many point out, is actually social democracy, a form of economic management practiced most notably in Scandinavian countries.
So if what I have said is true, you may ask: What is the purpose of the word "Communism"? The truth is that 19th century authors used the term "Socialism" and "Communism" interchangably; this can be seen in Marx and Engels, Bakunin and Oscar Wilde's works. Although they were separated into higher and lower stages, the practice of calling the lower stage as "Socialism" is an invention of Vladimir Lenin who sought to describe his country as "state Socialist" in an effort to convince people of the idea that the means of production were communally owned by the working class.
Communism is further not about party control; the idea of the vanguard party again originates from Lenin; but here we must make a distinction - Lenin did not seek to modify what his theoretical predecessors meant by "Communism", he sought to create a model of praxis, that is, to ask and answer the question of: How is Communism achieved? Lenin's own idea to this was the usage of the vanguard party, which is a group of highly educated Communist intellectuals which guides the masses of the working class toward revolution and Communism.
You are conflating Communism with praxis (thus making a category error) and further conflating that specific Leninist praxis with Communism in general. The evidence that this is a conflation rests in two facts: there exist and have existed through history democratic Socialists who not only used the terms "Socialism" and "Communism" interchangably as I have already mentioned but who sought to establish Communism not via representation of the working class themselves but via the normal methods of parliamentary democracy. An example of this praxis in use is various Socialist parties which compete in local and national elections in Europe and elsewhere. Further, there exist today several varieties of Communism, within academia the meaning of Communism can be stretched much father than you may have anticipated; Badiou writes, "where there is a State, there is Communism to oppose it". This is known as the Communist hypothesis.
On to the definiton of capitalism, the main factor is the private ownership of social means of production. By this I mean that the majority apparatus used to produce goods which society exchanges and uses are owned by individuals who seek to make a profit. This is echod by Smith, Ricardo, Marx and Keynes. Other factors which contribute to the definition include the predominant usage of wage labour and the goal of capital accumulation.