People use political labels pretty haphazardly, it is what it is. There’s no real right or wrong definitions, but we can aim for ones that at least cut reality at the joints and illuminate more than they obscure about the people they’re labeling. So here are my definitions:
Right-Left: To be on the right means to be in favor of concentrated wealth and power. To be on the left means to be in favor of dispersing wealth and power widely.
Progressive-Conservative: To be a progressive means to be part of a political movement to make positive change. To be a conservative means to be against whatever it is the current progressive movement wants to do. We can also add “reactionary” to this category, which is actually pretty similar to progressive except the change they want to make involves a reversion to a real or imagined past state of affairs.
Liberal-Authoritarian: To be liberal means to be tolerant and have a “live and let live” attitude. To be an authoritarian means to believe that there are correct and incorrect ways of doing things that need to be enforced.
I would consider myself a left-progressive-liberal. I’d say the modern Democratic Party are mostly centrist conservative liberals, while the modern Republican Party is composed of right-wing reactionary authoritarians. I call the Democrats conservatives because they’re for the most part just trying to defend the gains of past progressive ages rather than make any particular new ones. Even the new policy prescriptions they have are mostly about shoring up systems that are coming apart (healthcare anyone?) rather than embarking on some bold new plan to improve people’s lives.
The American culture wars are pretty much fought based on the liberal-authoritarian axis, on which the country has been steadily moving in the liberal direction for decades. And really that’s for the most part what elections are won and lost on, since those tribal culture war issues are what really seems to get people riled up. Meanwhile we’ve been moving to the right for a few decades despite there not really being much of a popular outcry for doing so because that’s what the right-reactionaries driving the Republican Party really care about.