Thoughts? Other ideas that do not fit neatly into the above, or are another (perhaps better) way of thinking of Bingo Card or Card Trick?
My view is that it boils down what is in-game and what is out-of-game.
The Railroad is what it is because the referee decided that there is only one way for the campaign to unfold for reasons other than that how the setting works. But a railroad as most define it is only one example of this. There are several ways of running campaigns that are ground on out-of-game considerations.
The Sandbox is what it is because the referee is fine with whatever the players decide as long it is something they can attempt as their character and works with the physics (or metaphysics if fantasy) of the setting.
The logistics of how to handle of all this is where things get muddled. The way to clarify things is to look at what is the primary focus of the campaign. The rest follows. For example I focus on letting players do whatever they want within the setting as their characters. In a bit of humor I call this "letting the players trash my setting." Because back in the day (circa 1980) when AD&D was all the rage among my high school friends. I knew several referees who got bent if a player tried to mess with any of the NPCs of their setting beyond the immediate adventure. My campaign stood out as something different because I just went along with the flow if the player decided that now is a good time to knock of the king. Usually it wasn't that grand but generally it was something that altered the landscape of my setting (Greyhawk then Wilderlands).
I was interested running my campaign this way because I started with hex and counter wargames first in the late 70s and then got into roleplaying. So I was used to playing out scenario where you were allowed to a lot of things to achieve the victory conditions. Then working with my friends in coming up with new scenarios to continue to play favorites. We some scenarios we tried were done to see what happen when we play under certain conditions.
That attitude got carried over to my D&D/AD&D campaigns. Once I found that players really like playing a new campaign set in the same setting where they made a change, it started me on the path that lead to what I write and referee today.
But if I didn't focused on that. Then what I would have done would have very different. Sometime that I learned when I tried other forms roleplaying like live-action in the 1990s. The boffer LARP I played, NERO, was basically D&D in the woods. But the logistic of running a live-action event meant that some elements far more flexible and natural than tabletop (players, NPCs staff roleplaying with each other). And other were far more restrictive (adventures and encounters). The basic problem is that staff and players need to rest, sleep and eat. That you can't have a adventure on one side of the site and then have another on the other side without building in the time and actually shuffling stuff around like props. So "adventures" are pretty much railroads except when they are mostly roleplaying. So I learned how to run railroaded adventures that players found fun and interesting to play.
It didn't affect my tabletop campaign because I still focused letting players trash my setting.
So the above is a long winded approach to the answer your question. If you want to know about the alternative, then the question to ask is what other things referee focus on for their campaigns other than events has to unfold "just so" (railroad) or a focus on letting players trash the setting as their characters (sandbox).
I will add that once the decision is made that player can do anything as their character in a setting (sandbox). There is not much in the way of subcategories. Everything afterwards is about logistic and how one manages the details of the setting to make the above happen. One sandbox campaign can wind up with the players deciding to be members of a military under orders. Thus one of the things the referee has to do is roleplay their commander giving out orders. Which at first glance doesn't appear much different than other types of campaign. While another sandbox campaign starts out the players on a blank hex grid with the freedom to explore in any direction.
Both are sandboxes because in both the players chose to be where they are at or start out at and they have the freedom to decide something different at any point because that what the referee has committed to focusing on. Of course there will be consequences to any decision. But one of the appeal of participating in a sandbox is the interplay of expected and unexpected consequences to decisions. In the campaign where the players choose to become part of the military, they could desert, they could roleplay in a way that they get an opportunity to resign, or they could opt to serve in the military throughout the duration of the campaign. Who knows what will happen and that part of the fun.
If the focus on something else the referee and group will find a different set of technique to be used. For example Railroads are neither good or bad. There is a way to run a good railroad, I find it harder than running a sandbox because the trick the same trick used in film or tv. You contrive a story that is interesting enough to make the audience want to keep sitting there watching. In the case of a railroaded adventure, the interesting story makes the players want to do the next encounter. The best make it so that the transition is nearly always seamless. Where to go for the next railroaded encounter is both obvious (but not too obvious) and compelling.
