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.
LAWYER, BATTERY, RHINO, and CARD all fit the category "Things that charge". Charge has a different meaning for each. 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 / MAP / HOTEL
PIANO, LOCK, MAP, and HOTEL 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
Words after moon
WALK / BEAM / STONE / LIGHT
WALK, BEAM, 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
Words before board
KEY / DASH / SURF / SCORE
KEY, DASH, SURF, and SCORE all fit the category "Words before board". Put one word after each answer. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.