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
Green things
MOSS / LIME / PEA / FERN
MOSS, LIME, PEA, and FERN all fit the category "Green things". They commonly show up in green. 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
DIAMOND / CLUB / SPADE / HEART
DIAMOND, CLUB, SPADE, and HEART 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
Photo terms
FOCUS / LENS / SHUTTER / FLASH
FOCUS, LENS, SHUTTER, 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.