> which bits of the Soviet projects were targetting making everyone equal
Private property of means of production and accumulation of capital were outlawed. It was the best attempt to follow Marx tenets.
> It strikes me that in all cases a portion of society wanted to impose communism, those who didn't want to give up their supremacy (of power or wealth) perverted that system.
No. The latter were massacred in 1918. Then the remaining ones executed and sent each other to GuLAGs in several waves. Separation of wealth (even if minuscule on the scale of developed capitalism) appears to be an emergent property.
That mainly seems to happen at the large scale. At the small, community scale, there are plenty of examples of communal societies with many decades of history that continue as an ongoing concern. For example, the Bruderhof's in New York and Pennsylvania:
https://en.wikipedia.org/wiki/Bruderhof_Communities
Genuine question, which bits of the Soviet projects were targetting making everyone equal -- all I've seen are the bits where a dictator rides the coattails of communist ideology to subdue the people. I'm still learning about this part of history, so am keen to understand which bits were started "properly" from a communist perspective. It strikes me that in all cases a portion of society wanted to impose communism, those who didn't want to give up their supremacy (of power or wealth) perverted that system.
As (IIRC) Galbraith says (paraphrase?) "in Capitalism man exploits man, in Communism it's the other way around". The thing is that communism at least seeks to gain equality, capitalism seeks to exploit others up to the legal limits imposed.