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.
DIAMOND, HEART, SPADE, and CLUB 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
Things with strings
VIOLIN / KITE / PUPPET / HOODIE
VIOLIN, KITE, PUPPET, and HOODIE all fit the category "Things with strings". A string is part of the object or how it works. 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
TOKEN / TILE / DIE / PAWN
TOKEN, TILE, DIE, and PAWN 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
Computer keys
SPACE / SHIFT / ESCAPE / ENTER
SPACE, SHIFT, ESCAPE, 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.