I've been spending the whole thread arguing against this dichotomy.
There is a type of game that is neither linear - in the sense that it has no defined path - but is not a sandbox. It is a game in which the GM presents the players (via their PCs) with situations that engage the known concerns of the players (signalled via PC backstories plus choices made in previous episodes of play) and then sees how the players respond.
Without knowing more about how you play and prep, I can't really say whether you are on an adventure path or in a sandbox. But the important point that has to be made is that I can't know which of the two you are doing because what you have described is (IMO) simply 'good DMing' and is a feature of either good linear or good sandbox play. Whether you are doing a sandbox, or whether you are constructing an adventure path from week to week, presumably the good DM is feeding the players situations that engauge their known concerns signalled via PC backstories plus choices made in previous episodes of play. I'd be doing that even if I was running the Dragon Lance moduels (as written, a very railroady adventure path) or if I was running an open ended 'pirate campaign' where the PC's sailed around in a boat exploring a world I had detailed.
There is of course more than one axis by which game structures can be plotted and described. Linear vs. non-linear is not the only feature of games, adventures, and campaigns. Just because you can pick out a salient feature of your game that doesn't lie on the llinear vs. non-linear axis doesn't mean that the distinction between an adventure path and a sandbox isn't real. It could be that you are playing in a fuzzy grey area in the middle where you switch back and forth between linear preperation for the session (tonight A will happen, then B will happen, then C will happen) and sandbox preperation (typically a map, location details, random encounter tables of various sorts whether events, monsters, or weather, large lists of wandering NPC's, etc.), and so the most salient feature of your prep is the tailoring you do to the dramatic conflict and themes proposed by the players.
The axis you are talking about, who controls the dramatic theme of play, is an interesting one - one that gets alot of discussion at Forge - but it doesn't mean that play isn't linear vs. non-linear. One additional way to describe a Railroad (the game type, as opposed to the act of railroading) is that it is not only linear, but the players have no ability through play to control or choose the dramatic conflict or the themes of the story. It is usually a feature of sandboxing that the players have greater control over the theme than in linear play, but its quite easy to imagine a Rowboat world where the players don't control the theme because there is no theme and they lack the ability to create meaningful dramatic conflict because the GM simply isn't putting enough effort into the story. Similarly, we could imagine a DM running the Dragon Lance modules with player created characters rather than pre-generated ones, and the DM tailoring the game to the conflicts generated by those party and intraparty conflicts rather than the stock ones presented by the modules and official story. The story is still on the whole linear with familiar places and events, if slightly different from the linear one presented in the modules, but the small events of the story revolve around different tensions and the PC's may have different presumed relationships with various NPC's. One of the original PC's might for example turn out to have been a dragon all along. Another might end up being a dragon Highlord's son, etc.
Lastly, I don't want to give the impression that I think good GMing means that the players are solely in control of the stories dramatic themes. The GM is a player too, and as the player most invested in the game (in terms of time and effort put into it, for example) the GM has a the reasonable expectation of having very large say what the story is about. Most players expect and have a reasonable expectation that the GM has a very interesting, exciting, twist filled, and thought provoking story in mind. Very little to me says poor DMing like a DM that doesn't have a story in mind and expects the players to effectively do all the work of world and story creation. I'm just saying that the good GM, in addition to crafting his own compelling story, takes cues from the players regarding what they are interested in.