A hex crawl is almost always Linear. But sure you can Sandbox any game.
Yes...okay...this is what is always said. So:
Okay a DM creates a massive game world.. So the game starts at 6PM. At 6:01 the players pick a goal or something to do. Okay....so....at 6:02 after the players have made their pick in the "the amazing sandbox", is the game STILL a sandbox? How? Why?
Like say the players look at the sandbox map and say "we go to the dungeon of Kassth. So the PCs travel to that location and go through a dungeon adventure. So is that all a "sandbox game" just as they players had the freedom for a second to pick "anything" to do?
Well, yes. The players were not presented with a specific direction; they chose their own.
As with so many things, the linear/sandbox descriptions are points in a spectrum, and gameplay takes place in the area between the two extremes. In even the most linear of TTRPG adventures, there is room for non-linearity, where players make choices and the DM improvises. In most sandbox campaigns, there will be plot elements, specific encounters, locations, factions etc that the DM has established ahead of time.
The "purest" sandbox campaign might be a hexcrawl with random encounters/wandering monsters and locations, with no actual plot, but to actually have an entire campaign of that would be pretty rare I believe.
What you are describing as "sandbox" sounds more like collaborative storytelling than anything that I'd describe as a conventional REG like D&D. While D&D does have collaborative storytelling, there is a structure that is not present in what you are describing.
Now of course you are very much entitled to have your own personal definitions of what some words mean, just like Kiyron45 does, that differ from the OP. But the OP did a pretty good job of defining the terms for the purpose of this thread. Why not engage with the discussion in that spirit rather than telling the creator that they are wording wrong because you word differently from them?