That's not how I see it.
The difference between the players can declare whatever actions they like for their PCs and the players are expected to declare actions that conform to a pre-established sequence of events is clear.
Correct. It is clear. One is playing a roleplaying game like D&D, the other like playing Blades. Both are roleplaying. No need to differentiate the roleplayers, because the differences are in the rules - not the players.
The difference between the GM secretly changes mechanical details like dice rolls and hp totals on the fly and the GM doesn't alter those details, and/or manages them in the open, is clear.
Yup. You are correct. And look, the DM can play D&D and be open at the same time. The definition of railroad did not insist on the DM altering dice rolls. It said they
may do this. Just like a GM in a Story Now game may try to do as well. Again, seems to me there is no difference other than a narrative that makes something look negative - even when worded to try and sound unbiased.
The difference between the GM creates new bits of backstory - second-stringers to replace defeated BBEGs, or clues to prompt the players to make the "right" action declarations - in order to keep play "on track", and the GM doesn't do that, is clear.
Oh look, a DM in D&D can do the latter. The can do the former. Many can cast mend, and combine the two, especially when they see the player's making decisions to continue down the path or forge a new one. And by
clues in quotes (my bad, clues was not in quotes, right was), you mean foreshadowing, setting dressings that create mood, and themes that orchestrate or highlight symbols. Those bastards!
The difference between the GM uses their authority over scene-framing to ensure that a series of pre-authored scenes take place and the GM frames scenes in accordance with some other principle - eg extrapolating from the prior backstory (as in a sandbox or map-and-key dungeon) or following player cues (as in Burning Wheel) or building on the fiction and the action declarations in a soft-then-hard-move pattern (as in AW or DW) - is clear.
You are right. They are all clear. One is called running an adventure path. The other is called running a sandbox. One is called a dungeon delve. All your definitions are correct. No need to cast comprehend language. And better yet, now you can disregard the terms railroading, force, and other things that aren't needed.
Who is confused about these differences? The controversy, as best I can tell, is around asserting that these difference might matter to someone's engagement with RPGing. Actually spelling out these differences, and asserting an unequivocal preference in respect of them, is taken to be some sort of tactless faux pas.
No one is confused at all. That is why this thread is 90 pages long.
The only confusion I see are the terms becoming so ambiguous, yet still being held onto because it is a philosophy some are not willing to part with.