In ancient Sparta the Spartans were called 'Homoioi'. This means 'equals'. They were called that way because they only received Spartan citizenship when they passed through the Agoge. This means that spartan citizens were regarded as equals through citizenship.
In ancient Athens every citizen had the right to the Agora go to the and have their voices heard. This means that on top of the citizenship equality, they also possessed political equality. Every citizen has the right to a political platform. This is the basis of democracy. 51% majority voting is the decision making mechanism that is only democratic when every citizen has a right to a political platform.
Places like the Soviet Union had 51% majority voting mechanism, but were not democratic. Why? Because citizens did not have political equality, meaning that they did not have a right to a political platform.
Before you say Sparta was not democratic, it was, its democratic body of government was called the Apella.
Egalitarianism is literally an entirely different ballpark. Egalitarianism asserts that:
"Egalitarianism is the doctrine of the equality of mankind and the desirability of political and economic and social equality."
Do you read that? Desirability of political, social and economic equality of entire mankind. Never before in the history of mankind has it been asserted, that all humans be created as equals. Neither that they should have access to an equal amount of money.
And you know why?
Beyond political equality and equality of citizenship, egalitarianism is an opinion.
You CAN NOT become economic equals. There is no maximum amount of wealth therefore it cannot be distributed equally, therefore there is no economic equality possible!
Social equality is saying that a drug user who abuses their own body is the equal of an elite athlete who treats their body with the respect it deserves.
If these two are indeed to be considered as equals it is an ideology of egalitarianism or as it says, the desire to equality.
Never again say that equality and egalitarianism are the same thing. The same fallacy is to think that wanting social policies is socialism because both are about 'social'. One is a tool the other an ideology.
Thank you very much for reading my rant all the way to the end! I hope it was at least a little informative.