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
Things you fold
MAP / LAUNDRY / ORIGAMI / LETTER
MAP, LAUNDRY, ORIGAMI, and LETTER all fit the category "Things you fold". A crease is part of the process. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Things that charge
RHINO / CARD / LAWYER / BATTERY
RHINO, CARD, LAWYER, and BATTERY 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
Board game pieces
PAWN / DIE / TILE / TOKEN
PAWN, DIE, TILE, 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.