You seem to have the truest take. It just boils down emancipation. Leftist policies are liberal in nature, seeking freedom from the hierarchy of capitalism.

Most of the confusion seems to come from neoliberalism or market liberalism co-opting the term for conservative and authoritarian uses and now everyone is confused.

Leftists who need a distinction in the terms seem to fall into two categories; those who see liberals as reformists instead of revolutionaries (too moderate), and the false leftism of Lenin/Stalin etc sects who aim for an authoritarian model but without capitalism.