Ovinomancer
No flips for you!
In referring to a "classic sandbox" with a "somewhat static, reactive character" I'm following [MENTION=16586]Campbell[/MENTION]'s post 65; and [MENTION=99817]chaochou[/MENTION]'s post 41:
The "scenarios" in this style are essentially "static" situations that the PCs engage via their PCs. The examples I have in mind, based on my own experience with material being published c1977-c1982, are for B/X, OD&D and early AD&D, RQ and Traveller.
The Village of Hommlet is an example that [MENTION=29398]Lanefan[/MENTION] and I discussed upthread - we were in agreement that the situation is "static" until the PCs inject themselves into it, which I regarded as a virtue but Lanefan as a weakness.
The classic dungeon is another example - be that a semi-serious dungeon, like the example of The Haunted Keep in Moldvay Basic, or a funhouse dungeon like White Plume Mountain, or some intermediate example like Moldvay's Castle Amber.
In the case of Traveller, I'm thinking of White Dwarf scenarios like The Sable Rose Affair or Amber to Red; or GDW modules like Mission on Mithril.
These scenarios don't have a trajectory of their own. They are situations conceived of by the referee for the players to engage via their PCs - poking here, asking questions there, gradually building up a picture of the situation so that (ultimately) they can "beat" it.
Because the situation is static but for the response to the PCs, the actual sequence of events in play is driven by player choices - they choose which rooms the PCs enter and which they ignore; they choose whether the PCs try to sneak past the guards or assault them; when these sorts of scenarios are incorporated into a larger "world", the players choose which "hits" to make and which to leave.
In this sort of play, a lot of NPC responses are determined randomly (reaction rolls; evasion rolls; etc - with the players being able to influence this by standard strategies like offering bribes) or by generic scripts (hobgoblins hate elves and always attack them; skeletons fight until destroyed; etc) which the players are capable of learning via divination magic, collecting lore from NPCs (think of classic D&D's elaborate rules for sages), etc.
When the GM turns the "world" into a "living, breathing one" - ie the sorts of changes [MENTION=29398]Lanefan[/MENTION], upthread, said that he would prefer to see made to T1 - it is no longer the players who are choosing what parts of the "world" to engage, and driving the fiction by their choices. The world is going to come to them (eg if the players ignore the orcs, the GM works out orcish events offscreen, and these then feed back into the events that occur to the players). The extreme example of this is the players whose PCs ignore the cultist plot and find, X amount of game time down the track, that the world has come to an end in a great apocalypse. But the same trajectory of play can unfold in less extreme cases.
I didn't see anything in Lanefan's description that violates your definition, though -- he presented various things going on that the players then can choose to engage with.
As for the 'sandbox in motion' idea, you using a narrow definition that requires a static world, only changing due to player engagement, is harmful to full discussion. A sandbox, generally, is a place where players choose what to engage -- the world is open to player engagement. Nothing about this requires that the box not move when the players aren't there. Your example of a cult ending the world behind the scenes is using a bad example to dismiss an idea -- having a game end without any player engagement in the reasons is just bad GMing, in any system or method. But having the players have to deal with complications for things they knew about and ignored isn't -- ie, if they learn about the cult and decide to go knit sweaters, having the town they knit in taken over by summoned demons isn't a failed sandbox. The concept of Fronts, above, largely mirrors the ideas I have for a living sandbox. A world that doesn't change unless a player looks at it is boring.