OK, I see what you mean. Does that correspond to actual political parties in the UK? Is there a "liberal" party that is considered neither left nor right?
I would say that in the US, what you're calling liberal would translate to "centrist/mainstream Democrats" who believe in the market but also in a social safety net. In other words, the majority of Democrats. But we also just call that the left, because it's the mainstream political viewpoint that is opposite to the right.
What you are calling the left, we call "progressive", which is why you see so many references to the "progressive wing" of the Democrats. Which includes Bernie and also AOC, who also call themselves "socialists", but in the US this doesn't mean communist -- it's not about government ownership, but vastly stronger regulation, protection, and government action generally.
While Elizabeth Warren is really her own idiosyncratic category. She's doing her own thing that isn't really aligned with mainstream Democrats or with the progressives, or with anybody else particularly. If anything, you might call her more "technocrat" than anything else.
But at the end of the day I hope I've answered your question as to why liberal = left in the US. Because liberal means pro-equality, and for whatever historical reasons, equality moved from mere legal equality to a more robust equality of opportunity. And we use "classical liberal" to distinguish the old liberal from the new.