BotE, I agreed with most of your post but wanted to quibble with this:
If the GM sets up the scenario/situation, but is doing so based on the cues provided by the player(s) through character build, character background and prior actual play, then I don't think this is a railroad at all. It's just the GM doing his/her job in a non-exploration focused game.The most useful definition for sandbox I've heard is something along the lines of, "Allowing players to choose the scenario." The opposite of that is, "Not allowing the players to choose the scenario." That's certainly a kind of railroading, but it's pretty much the lightest form of railroading possible.