The Earth's gravity well extends far beyond the atmosphere. The moon is for example is well inside the gravity well and even the hill sphere meaning Earth's gravity is the dominant gravity force on the moon.
I'm talking out my ass here but I think outer space literally meant "outer area" as in the area outside of our planet, and we're so used to that term that it's turned into the proper noun Space. Earth (or whatever celestial body is your current frame of reference) is implied to be the inner space.
From one of my favorite episodes of X-Files, where I first heard the term "inner space," I define it as subterranean/oceanic. The space within our planet. The episode was talking about aliens that came from the center of the Earth, as opposed to space. And I always thought that was a neat twist. It was also one of those weird ass ones Duchovney directed so it was high on the humor scale and featured the Lone Gunmen which I always liked.