These are just fan-terms.
MMO's basically fall into three camps
"Themepark" which is where every piece of content is delineated like a "theme park" and whatever you do in A has no bearing on B. eg you can play the storyline or skip it, and the world will remain the same. Likewise what you do in the game has no bearing on what someone elses storyline.
"Sandbox" which is where you have a lot more control over the world beyond just making a house in a "housing area", sandbox MMO's do not use instances or shards to create multiple versions of the world (for more players), rather they have one world that isn't indelible, the content reflect it. So far no MMO in existence actually qualifies for "sandbox" because ... The closest thing are minecraft clones, but they all fail the sharding test. Sandbox MMO's are designed for Role-play by not providing a thematic element.
"Pseudo-sandbox", which is somewhere between themepark and sandbox. These are games that adopt some sandbox-like elements (eg placeable housing, large explorable areas, physics engines) in a larger themepark style MMO. Most games that call themselves sandbox MMO's are actually this. So the players can have some soft-effect on the world (that other players can undo) but in the end, if a player quits the game, everything they left in the game will disappear or be claimed by someone else.
The point is that a "themepark" mmo is designed around storyline content. You can have multiple storylines. You don't have to pay attention to the storyline, but there is a "theme" running through all the "rides" (eg dungeon content, overworld content, crafting, etc)
Other kinds of games (eg Destiny, PSO2) that have some dynamic multiplayer elements, but are otherwise single player games with one multiplayer lobby are also just themepark MMO's.
A "MMO" itself is "Massive Multiplayer Online" and can refer to any game with an online component, including games like farmville, even if all the game does is invite your friends for goodies. However this classification of game is neither a themepark or a sandbox, because the world isn't shared at all, and the extent of MMO'ness is visiting another friends town. Just like minecraft.
So generally all MMORPG's are Themepark MMORPG's even if they call themselves sandbox MMORPG's. There are no true sandbox MMORPG's, and at best, the sandbox elements are just another "theme" to the themepark, that is completely optional play with.