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, COVER, INDEX, and CHAPTER 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
Hidden numbers
FREIGHT / HEIGHT / TENSION / STONE
FREIGHT, HEIGHT, TENSION, and STONE all fit the category "Hidden numbers". Each word hides a number sound or spelling. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Garden tools
SPADE / RAKE / HOSE / TROWEL
SPADE, RAKE, HOSE, and TROWEL all fit the category "Garden tools". These live in a shed. 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 fly
JET / EAGLE / DRONE / KITE
JET, EAGLE, DRONE, and KITE all fit the category "Things that fly". They spend time in the air. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.