The reason that I'm optimizing those sentences away is, they aren't pulled back into the central question. I take as my central question: why is walking better than sitting for some kinds of intellectual problems? That first quote doesn't connect to that question (it says that there may be differences between different sorts of locomotion), nor does the second quote (which doesn't distinguish walking from sitting), nor does the third (it tries to distinguish city-walking from garden-walking but not walking from sitting). Heart rate was the only thing I saw which actually seemed to answer that question of, "why does walking help us think?" as opposed to "how should I walk, so as to think better?"
But my point was more, "I was hoping for some science," really.
My personal hunch is that, at least for me, the benefit stems from constantly-changing scenery. I'm not actually paying attention to it most of the time, but little bits catch me here and there, and in any case I'm aware of it. This passive, non-urgent sensory stimulation doesn't make me forget what I'm thinking about, but constantly interrupts my inward fixation just enough that I don't get stuck on one thought -- so new ones have a crack to wedge themselves in through.
It could boil down to an instinctive awareness that I'm not in a place I know is absolutely safe, so I have to be ready to accept and react to external stimuli at any moment, even if I'm not currently feeling threatened; and creative thought, in this case, hijacks the "external" tag.
A moot point. The scientific studies ruled out treadmills as an effective alternative. Your first post make it appear as if you hadn't even read the article. I'd argue that increased heart rate was demonstrated not to be a likely cause.
But my point was more, "I was hoping for some science," really.