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.
BEAM, WALK, STONE, 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
Stage terms
PROP / CURTAIN / SCENE / ENCORE
PROP, CURTAIN, SCENE, and ENCORE all fit the category "Stage terms". A theater audience might notice them. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Airport words
RUNWAY / LUGGAGE / BOARDING / GATE
RUNWAY, LUGGAGE, BOARDING, and GATE all fit the category "Airport words". Travel days are full 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
Things in a gym
BENCH / KETTLEBELL / ROPE / MAT
BENCH, KETTLEBELL, ROPE, and MAT all fit the category "Things in a gym". Training rooms are their habitat. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.