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.
WALK, STONE, BEAM, and LIGHT 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.
Category
Baking staples
BUTTER / SUGAR / YEAST / FLOUR
BUTTER, SUGAR, YEAST, and FLOUR all fit the category "Baking staples". A bakery keeps these close. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Map features
LEGEND / BORDER / COMPASS / SCALE
LEGEND, BORDER, COMPASS, and SCALE all fit the category "Map features". These help you read a map. 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
COIN / MOUSE / COMET / KITE
COIN, MOUSE, COMET, and KITE 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.