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.
RUNWAY, BOARDING, LUGGAGE, 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
Computer keys
SPACE / ESCAPE / ENTER / SHIFT
SPACE, ESCAPE, ENTER, and SHIFT all fit the category "Computer keys". They are labeled on many keyboards. 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 bars
CHOCOLATE / SIGNAL / SOAP / MUSIC
CHOCOLATE, SIGNAL, SOAP, and MUSIC all fit the category "Things with bars". Bars can describe 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 after moon
LIGHT / BEAM / STONE / WALK
LIGHT, BEAM, STONE, and WALK 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.