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, 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
Photo terms
SHUTTER / LENS / FOCUS / FLASH
SHUTTER, LENS, FOCUS, and FLASH all fit the category "Photo terms". A camera owner knows these. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Soft things
PILLOW / FLEECE / VELVET / MIST
PILLOW, FLEECE, VELVET, and MIST all fit the category "Soft things". They suggest a gentle texture. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Book parts
INDEX / COVER / CHAPTER / SPINE
INDEX, COVER, CHAPTER, and SPINE all fit the category "Book parts". They help make up a printed book. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.