Nitpick re "edge of space": Low earth orbit starts at around 160 km and orbits decay rapidly below 200 km. This is 36 km altitude. But what I think most people miss is that to "be in space" the way most people think about space you'll need to enter orbit which means that once you get up to 160+ km altitude you need to go sideways at roughly 24 times the speed of sound - and then you're in orbit. Then when you're all done having your fun you need to take all that orbital energy and do something with it. The best idea so far is to use it to burn up a heat shield. This business of going vertical a little higher than jets and a lot lower than satellites is a little pretentious. But it's fun and it's Red Bull so who cares!!
I'm not aware of any organization that determines being "in space" by velocity -- the FAI as well as the US DoD, NASA, and the FAA all go solely by altitude.
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.
