The main disconnect I see around this is people looking at an objective scale of political science versus a more contextual scale, like narrowing down to Western politics. In terms of an absolute scale, we're all authoritarians because we believe in having a centralized government. That classification isn't remotely useful however. Western politics tends towards capitalism and authoritarianism, and so it makes sense to discuss it with adjusted scales.

The axis is authoritarian-libertarian, not liberal. The definition might be different in common parlance, but people not understanding terms in political science through ignorance is not a reason not to use them.

A liberal socialist is a subset of liberals, the same as social democrats and social liberalism.

You cannot seek to preserve capitalism and also be a socialist.