I might not just be doing a good job offering a definition. That doesn't mean there isn't one. Because I can certainly recognize a sandbox when I am in one, when I see one, or when I am running one, versus an investigation, an adventure path, etc. I don't remember exactly how I phrase it but it was probably something along the lines of players have full freedom to explore and do what they want, to set their own agenda, to not engage 'the adventure' to have an impact on the setting, etc. In a linear adventure the whole point is there is an adventure to be had, and players expect to find the threads to that adventure and stay on the path. It is also expected to have a structure to it that is discernible (a set of encounters that all make sense and lead somewhere, a set of events that get more dramatic etc). A sandbox is much more open, but also a lot less concerned with those structural expectations. So the upside is the players can indeed go where they want, set out on any adventure they want, engage any element of the setting they want. If they suddenly decide to start a haberdashery empire then that is normal and expected in a sandbox. That isn't normal and expected in a linear adventure. And if linear adventure opens up enough that players are free to do all these things without any conceit of play constraining them, then they are most likely in a sandbox.
