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.
DIE, TILE, PAWN, and TOKEN all fit the category "Board game pieces". You might move or roll these on a tabletop. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Airport words
LUGGAGE / GATE / RUNWAY / BOARDING
LUGGAGE, GATE, RUNWAY, 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
Computer keys
ESCAPE / SHIFT / SPACE / ENTER
ESCAPE, SHIFT, SPACE, and ENTER 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
Desk items
LAMP / NOTEBOOK / STAPLER / PENCIL
LAMP, NOTEBOOK, STAPLER, and PENCIL 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.