People seem to have misunderstood yesterday's post, or misconstrued it, or not read it properly. Alternatively, it was poorly worded, although personally I doubt that. Here's something that will simplify. May I present:
No? I'll explain. On the left hand side of the spectrum are campaigns where the DM is totally reactive. This can be synonymous with player freedom, although this is a somewhat grey area and the mapping is not perfect. It is certainly synonymous with "front loaded prep" (i.e., the further left you go, the more the prep is weighted towards the period before the campaign actually kicks off).
The further you go towards the left, the more the players are driving the game; they choose what they want to do and where they want to go, and the DM just reacts to them. At the extreme left are the most sandboxy sandbox games, often using D&D or similar: the DM creates the world, or an area of the world, and it sits waiting for the players to appear and start doing whatever they want to do - going down dungeons, hexcrawling, interacting with the NPCs, whatever. (A lot of story games are also at the extreme left.) Another way of putting it is that here, the way events unfold tends to originate in the players. The dungeon does not come to them; they go to the dungeon. The hexes do not come to them; they go to the hexes. (In Soviet Russia, hex crawls on YOU.)
The further you move towards the right, the more the DM is driving the game and the more the players are reacting to it. This does not mean that the further you go towards the right the more the game resembles a rail road: rail roads do not feature on the the spectrum because they are not actually games.
Here, the players will have freedom, but their freedom tends to be constrained by what goes on in the world - in particular, it is constrained by the fact that events do not have an origin internal to the players, but external to them.That is, the further right you go, the less the players are the engine and the more they cede initiative. A superhero game would tend to be more towards the right, even if notionally it is "sandboxy" and there is no predetermined plot or end point; this is because superheroes tend to have to react to conflict that has an external origin. Superman does not hatch plots to fuck over Lex Luthor - the action originates externally to him. He has the freedom to foil Lex's plots in whatever fashion he likes, but still, he is not the one with the initiative.
The Samurai Sandbox leans towards the right. It is not all the way to the right; I would put it just slightly to the left of the "players partially reactive" line. This is because, in the Samurai Sandbox game, even if the players have very vague, very open, very ill-defined mission statements, the conflict they have to deal with is still largely of external origin.
To illustrate, let's use my "keep the peace in Tosa province" mission. This is a very broad goal which can be interpreted in any manner of ways, and certainly won't result in DM rail roading. But nevertheless, the game is still going to be one in which the players are reacting to events, conflicts and circumstances which originate externally. This is because you can't keep the peace if there is nothing threatening it (well, you can do, but it would be a rather dull affair); and because the players are the ones keep the peace, this means the threats have to have an external origin (bandits, wolves, ghosts, spirits, yakuza, Mongolian invaders, internal strife, whatever). The players are going to be in large part reactive.
They are not totally reactive of course - and any group of players worth their salt will be partially active and will still drive the game along in many other ways with their plots and schemes. But still, the majority of what happens in the game is going to arise outside of player action.
There is nothing wrong with this, of course, and there is no value judgement on Noisms' Reactivity Spectrum. However, as I alluded to at the beginning of the entry, the further right you go on the spectrum the more prep is likely to be evenly distributed from session to session, and the more prep is going to be involved. Not only does the DM have to create the set up before the campaign begins. He also has to keep coming up with many, and varied, threats, and he has to make the threats change and respond correspondingly as the players react to them. The campaign may be very fun, fulfilling, exciting and all the rest, but it is going to require considerably more work than the "pure" rogueish sandbox will to run.