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.
CLUB, HEART, 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.
Category
Things that ring
PHONE / ALARM / CASHIER / BELL
PHONE, ALARM, CASHIER, and BELL all fit the category "Things that ring". Each can make a ringing sound. 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
TENSION / STONE / FREIGHT / HEIGHT
TENSION, STONE, FREIGHT, and HEIGHT 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
Things in a gym
BENCH / MAT / ROPE / KETTLEBELL
BENCH, MAT, ROPE, and KETTLEBELL all fit the category "Things in a gym". Training rooms are their habitat. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.