Classical liberalism / neoliberalism are not left-wing ideologies. They are, at best, centre-right if the Overton window is set to Western standards.
Edit: I love being downvoted because I care about the meaning of words. I clearly stepped on some American nerves. Thanks HN.
The whole reason you think the meaning depends on the context is because you have been mislead to believe that liberal == left-wing, or something approaching that.
This is completely and utterly false, hence my "nitpick". Changing the meaning of these words causes the exact type of confusion you seem to be suffering from!
Left-wing is a term with a very loose definition, and depending on the Overton window, "liberal" places somewhere on the left/right axis. "Liberal", however, does have a well-defined meaning, since it refers to a specific ideology, unlike "the left".
TLRD: "liberal" is not a direction on an axis, rather it's a point on an already existing axis, the left/right one.
Interestingly, you see this most in Australia, where the Centre/Right party is called the Liberal Party.
But it hasn't been used like that since the 1970s really, except in the "Liberal Democracy" sense. I think that has actually caused a lot of issues where the US right says they don't support Liberal Democracy, whereas - generally speaking, with the exception of the authoritarian arm - they mostly do.