I'm not an academically trained scholar regarding left-wing theory, but I'd assume that communists and social democrats are still part of the same group, with one naming themselves after a shorter-term goal-state, and the other naming themselves after a longer-term goal-state.
When we talk about state models such as republic, democracy, autocracy, we're either describing a current status, or a model we might want to follow or avoid. When we talk about ideologies (conservatism, liberalism, communism, feminism, etc.) they assert specific values and presumptions that might or might not be true or workable. For instance, in the communist ideal, every participant has exactly the same amount of political and material power; influence is perfectly distributed. But we have no idea how a state like that would look, or work, or if we could ever get there.
Every model and every ideology has problems and concessions we don't understand and have to correct for. The one-person = one-vote thing seems intuitive for democracy, but has terrible side effects, and we're still sorting out alternative election models that might work better.
All this is to say it's a really bad idea to treat any one of them as a racehorse or football team or a banner under which to rally and consolidate political power. None of the models or ideals we have are perfect or absolute, and we have to be prepared to adjust them on the fly, especially as we content with corruption and bad actors who exploit vulnerabilities.
I suspect everyone on the left ultimately seeks a society in which everyone is materially provided for, in which liberties are as extensive as possible while providing for protections and considering human biases towards certain abberant behavior (e.g. drunk driving) in which there are as few social strata as possible and power is as well distributed as possible. The models that accommodate all these, even to partial degrees, are still very fuzzy. (Western civilization has been working on them for only three hundred years or so.)
So we're at least in the same book, if not on the same page.