Sounds like you're just applying somewhat arbitrary purity rules to what falls into the "leftist" bucket or not.
I believe that for most people "leftists" means "on the left side of the political spectrum", and so, in the US, it would be a strict superset of democrats & Liberals. But for you, it sounds like it's only subset of these, based on some criteria you haven't made explicit.
Fair enough, but that's a somewhat non-standard definition.
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.
It's standard among self-described leftists, to distinguish themselves from establishment liberals.
I'm not even very far left and I identify with the term because party leadership are all 80yo and out of touch.