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.
SPADE, DIAMOND, HEART, and CLUB all fit the category "Card suits and kin". A deck has all four. 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 / HOTEL / MAP / LOCK
PIANO, HOTEL, MAP, and LOCK 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
Things with strings
VIOLIN / PUPPET / HOODIE / KITE
VIOLIN, PUPPET, HOODIE, and KITE all fit the category "Things with strings". A string is part of the object or how it works. 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
BEAM / STONE / WALK / LIGHT
BEAM, STONE, WALK, 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.