"Leftism is defined by a goal of communism or socialism"—but that is just communism. No other ideology seeks communism as a goal, and socialism is just a stepping stone to communism, or at least that is what they taught me in my Soviet high school.
You ever stop and think that the authoritarians may have been lying to you?
Outside of communist thought (wherein socialism is a stage on the route to communism—and the only concrete stage to which there is a roadmap in most communist thought, though different schools of communism have different maps), socialism is a thing sought in its own right by socialists who are not communists.
Obviously, its unsurprising that a Soviet high school would only teach the dogma of the particular brand of communism then currently held to be orthodox in the USSR, but that's a rather limited view.
I'm sure there's better definitions too, but neoliberals are not a leftist in any of them.