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.
TOKEN, DIE, TILE, 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.
Category
Night sky
METEOR / VENUS / NEBULA / ORION
METEOR, VENUS, NEBULA, and ORION 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
Kitchen verbs
FOLD / SEAR / DICE / WHISK
FOLD, SEAR, DICE, and WHISK all fit the category "Kitchen verbs". These are actions from a recipe. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Things with crowns
WATCH / TREE / TOOTH / KING
WATCH, TREE, TOOTH, and KING 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.