Forgive me, but more and more I see the phrase 'outer space' referring to stuff that happens at distances between the Earth and the moon. Is this a thing now?
It's always been a thing. It's, for instance, the sense used in the 1967 Outer Space Treaty.
“Outer space” is basically everything outside of some fuzzy demarcation line (often, specifically, the Kármán line) marking the “edge” of Earth's atmosphere.
"Space" is just a shortening of "outer space" in this case. This is opposed to the "inner space", namely the space within Earth's atmosphere. Everything outside of that is "outer space".
It's a slightly older form of the term. "Space" is just space, it's everywhere. Your car has space inside it, so does your backpack, so does your living room, it's all just ordinary space. "Outer" space has specifically referred to the space above Earth's atmosphere, to differentiate it from more Earthly kinds of spaces. Over time outer space has become less exotic and so it tends to just be called "space" colloquially these days, with the meaning understood based on context.
“Outer space” is basically everything outside of some fuzzy demarcation line (often, specifically, the Kármán line) marking the “edge” of Earth's atmosphere.