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.
METEOR, NEBULA, VENUS, 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
Things with tails
MOUSE / KITE / COMET / COIN
MOUSE, KITE, COMET, and COIN all fit the category "Things with tails". Each can have or refer to a tail. 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
TROWEL / RAKE / HOSE / SPADE
TROWEL, RAKE, HOSE, and SPADE 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
Words after moon
WALK / BEAM / LIGHT / STONE
WALK, BEAM, LIGHT, and STONE all fit the category "Words after moon". Put one word before each answer. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.