It's a sandbox if the DM doesn't have a series of steps the players must accomplish in order to achieve their goal. My example was a request for help from a town with multiple NPCs including enemies and allies along with some monsters. I have no expectation that the characters are going to do anything other than presumably talk to the person that was asking for help. After that it's up to them. For that matter, if they get to the town and decide to do something completely different (i.e. the player who decided his character was bored so he just went for a walk) I will start improvising using the existing lore and possible monster groups I had created.
So the problem here is this is just a normal game. So what makes it a Sandbox?
They can decide to talk to the goblin king, infiltrate and try to avoid a fight, go in spells blazing. They could get there and decide to set up an ambush or any other of other activities. Admittedly I don't do "dungeons" per se, if they are going to the ruins of Kassth it's because there's some threat or mystery and I'll have the place populated. How the group interacts with that population is up to them. What won't happen is a series of planned encounters that must be accomplished in order for them to succeed.
Again, this is a normal game. What makes it a Sandbox?
As others have stated, I don't understand the hostility to the concept of a sandbox. I see a clear difference from what I do and linear campaigns. I don't plan encounters or results of encounters, I plan locales and inhabitants. I have no long term plots even if the NPCs and groups have long term plans of their own, how the players interact with those plans is completely up to the players. What the players do is up to them every step of the way.
And yet again this is a normal game.
There are only two points:
1.The hate for anything pre made by the DM as many players hate the power the DM has to create the gameworld
2.The players desire to "do nearly anything" with no limits, and for sure no limits made up by the DM-player
So the DM makes a setting, then steps back and lets the players choose anything they want to do....and everyone class this "a sandbox game"......I guess for the couple minutes the players choose.
So if a DM makes a location "the cabin of the retired ranger Roz". And this is the important point: does the DM make up any details or does the DM leave the details mostly blank?
See, if the DM writes down....say 12 paragraphs about Roz, this limits what the players can have their characters do. If paragraph 10 says "Roz is retired and won't join with the characters to help them", there are a huge number of players that hate that. They don't want the DM making such things in the game.
The players want the freedom of an Improv game to "do nearly anything", so they want the cabin of Roz to be 'blank' so they can add to the fiction and do whatever they want without the DM saying "no, it is written".