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.
SPINE, CHAPTER, COVER, and INDEX 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.
Category
Film crew roles
EDITOR / PRODUCER / DIRECTOR / GAFFER
EDITOR, PRODUCER, DIRECTOR, and GAFFER all fit the category "Film crew roles". They appear in movie credits. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
River words
DELTA / RAPIDS / BANK / CURRENT
DELTA, RAPIDS, BANK, and CURRENT all fit the category "River words". Fresh water flows through this set. 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
HEART / CLUB / SPADE / DIAMOND
HEART, CLUB, SPADE, and DIAMOND 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.