No, that's not what “leftist" generally means, it means adherence to any of a broad grouping of anti-capitalist ideologies (the more widely recognizable being the various forms of socialism and communism). It overlaps a little bit with the progressive wing of the Democratic Party in the US, but beyond that is mostly outside of the US major party system. The only people who use “leftist" to describe a superset of the Democratic Party also use “socialist” and “communist” in the same way.
I think you're injecting your personal views into the definition, which don't align with the common meaning of the word.
Here's the definition of the word taken from macOS's built-in dictionary (New Oxford American Dictionary):
> leftist, noun: a person who has left-wing political views or supports left-wing policies.
> left-wing, noun: the section of a political party or system that advocates for greater social and economic equality, and typically favors socially liberal ideas; the liberal or progressive group or section
Anti-capitalist thought represents a subset, but is certainly not the sole marker of left-wing political thought.
Now, my _personal_ take on this is that western left-wing thinking and liberalism (in the moral philosophical sense) are deeply intertwined (given the Enlightenment and its values), but anti-capitalist thought is deeply illiberal in nature. It is this fundamental contradiction that leads to permanent infighting within the left-wing spectrum.
> I think you're injecting your personal views into the definition,
I think I'm injecting what I know from getting a degree in political science and a two and half subsequent decades of experience paying attention to how political terminology is used by scholarly, journalistic, popular, and activist sources across the political spectrum into the definition.
> Here's the definition of the word taken from macOS's built-in dictionary
And that's a tolerably decent definition for a relatively compact general dictionary, but it misses a lot of nuance; outside of activist sources using it as a slur for their opponents, it is not used in nearly the local-politics-relevant way that left/right (especially qualified with a national label, like "American left") is.