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.
LUGGAGE, RUNWAY, GATE, and BOARDING 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
Stage terms
CURTAIN / PROP / SCENE / ENCORE
CURTAIN, PROP, 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
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
Coffee shop orders
AMERICANO / CORTADO / MOCHA / LATTE
AMERICANO, CORTADO, MOCHA, and LATTE all fit the category "Coffee shop orders". You can hear these at a counter before 9 a.m. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.