Tatamibari
Cut the grid into rectangles so that every rectangle holds exactly one clue, and the clue says what shape it is: + a square, − wider than tall, | taller than wide. And one rule that belongs to no rectangle in particular: no four rectangles may meet at a point. Drag across the board to draw a rectangle; click one you have drawn to take it back.
Three rules are a matrix; the fourth is not
"Every cell belongs to exactly one rectangle" and "every clue gets exactly one rectangle, of the shape it asks for" is exact cover, written out with nothing left over. Build one column per cell, one row per candidate rectangle, and hand it to dancing links. The clues do not even need columns of their own: every candidate contains exactly one clue cell, so the column for that cell already forces one rectangle per clue.
The fourth rule does not go in. It is not about a cell being covered once — it is about a lattice point being cornered at most three times. Exact cover speaks "exactly one" and, with secondary columns, "at most one". It has no way to say "at most three". So the rule rides along the search as a counter that vetoes a row after the matrix has already offered it.
And it is the rule that makes the answer unique
Take boards with exactly one solution, switch that single rule off, and count again. At 10×10 almost none of them stay unique — the matrix on its own is happy to accept several tilings, and the thing that picks one out is precisely the constraint that would not fit inside it.
Most tilings are dead before a clue is written
Draw a random rectangle tiling of a 10×10 grid with no piece longer than four and it breaks the corner rule 95% of the time. Cap the pieces at 2×2 instead and it is 100%, at every board size — four small rectangles meeting at a point is what a fine tiling does. The generator here draws tilings and throws almost all of them away.
A square clue is worth four of the others
+ pins both dimensions at once; − and | only put them in order. On a 10×10 board a square clue keeps about four candidate rectangles and a wide one keeps about fifteen. Same symbol size, four times the information.