The first stage (a walk) is a reflection of a profound realty: "I'm going for a walk"
The second stage (a hike) masks and denatures a profound realty: "I'm going for a walk and it's special because of the settler myth of 'the wild'
The third stage (an "urban hike") masks the absence of a profound realty: "I'm going for a walk and it feels special like a hike but it doesn't count as such because it's not in 'the wild'"
The final stage (trail running) has no relation to any reality whatsoever: "I'm going to run in the woods and bother people for 'fitness' (yeah right)"
i think of hiking as something done over uneven ground, maybe a dirt trail at most. a "walk" is a broader term, but is tended to be associated as a means of travel ("take a" bus, car, bike, walk) over a paved surface in the context of the built environment.
i don't assume this is technically correct, but if i am trying to talk plainly that's how i use them. confusing or blurring the boundaries can be an easy bit, because hike implies more distance and maybe even gear. "the bathroom is downstairs. it's kind of a hike."
also there's the dismissive go away... "take a walk, buddy" vs "take a hike, buddy"... the hike one seems to tell someone to go farther away, maybe in a direction away from other people even."
I'd say this is accurate, it's mostly how well-groomed the path is. A hike will require you to navigate over natural obstacles like rocks, roots, logs, streams, etc. at some point, where a walk does not. A paved path all the way up a mountain is a walk, but a relatively flat but rocky trail through the woods is a hike.
Hiking is to walking as mountain biking is to... regular biking. It'd be odd to say you were mountain biking if you just went around the neighborhood. Honestly even calling a walk through a field a hike is pushing it, but I guess that's as wild as nature gets in some places
As someone who loves hiking. Parks and such perpetuate this obnoxious settler notion that "nature" is a place separate from us, and that we are simply "visitors"to it. Rather than a part of a vast, dialectical, web of interconnected relationships.
I just call it an urban hike. It helps that one of the neighborhoods in my city is up a huge hill, but I'd still call it a hike if I was out walking all day.
Where I am a hike has to be in the mountains, anything else just wouldn’t sound right. But, maybe that’s a product of growing up by a huge mountain range that people frequently traverse recreationally. Hike to me implies a large change in elevation. Hiking up, hiking back down, etc.