What “solved” means here
A puzzle is solved when the board is finished and the rules hold. Every cell that needs a mark has one. No rule is broken.
Opening a game does not count. A board you left halfway is not solved. Restarting does not make the old attempt solved. Only a finished, legal grid is.
Solved is an ordinary word here. It means you finished this puzzle, in this browser, and the grid is legal. It is not a public badge. It is not a score.
When a board is solved, it can appear on the Solved list. Until then it can appear as Continue on Home, if you left it unfinished.