The full solution is below, with a short explanation for each group. To avoid spoilers, play the puzzle or open the hints page before reading the answer panels.
WATCH, KING, TOOTH, and TREE all fit the category "Things with crowns". A crown can sit on each of these. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Garden tools
RAKE / SPADE / TROWEL / HOSE
RAKE, SPADE, TROWEL, and HOSE all fit the category "Garden tools". These live in a shed. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Night sky
VENUS / ORION / NEBULA / METEOR
VENUS, ORION, NEBULA, and METEOR all fit the category "Night sky". Look up after sunset. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Board game pieces
TOKEN / TILE / DIE / PAWN
TOKEN, TILE, DIE, and PAWN all fit the category "Board game pieces". You might move or roll these on a tabletop. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.