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.
GATE, LUGGAGE, BOARDING, and RUNWAY 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
Desert words
MIRAGE / CACTUS / DUNE / OASIS
MIRAGE, CACTUS, DUNE, and OASIS all fit the category "Desert words". Think sand and heat. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Card suits and kin
CLUB / SPADE / HEART / DIAMOND
CLUB, SPADE, HEART, and DIAMOND 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
Desk items
NOTEBOOK / PENCIL / STAPLER / LAMP
NOTEBOOK, PENCIL, STAPLER, and LAMP all fit the category "Desk items". They belong within arm's reach at work. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.