I'm also not sure what you mean by "the way most people think about space". The first American in space, Alan Shepard, flew a suborbital mission. I don't think you'll find many people who would argue he wasn't in space.
To be clear, there is no universally-agreed upon definition of space, let alone "the edge of space". There is no official definition of space in US law or policy. The NASA, the FAA, and the Air Force definitions (100km, 100km, 50 miles respectively) are only used for the purposes of awarding astronaut wings.
Personally, I'm fine with them calling it the edge of space to get more public attention to the mission, and they can get away with it because the cameras will show black sky and the curvature of the earth.
Makes me interested to see what the treaties say as I'd be surprised it no one thought to limit the control of countries to allow satellites to pass without permission or allow space vehicles in general.
Though as you point out there is the Karman line convention.