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
Film crew roles
DIRECTOR / EDITOR / GAFFER / PRODUCER
DIRECTOR, EDITOR, GAFFER, and PRODUCER all fit the category "Film crew roles". They appear in movie credits. 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 keys
PIANO / LOCK / HOTEL / MAP
PIANO, LOCK, HOTEL, and MAP all fit the category "Things with keys". Each one can be associated with keys. 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
CURTAIN / SCENE / PROP / ENCORE
CURTAIN, SCENE, PROP, 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.