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.
PEPPER, GINGER, CHILI, and WASABI all fit the category "Spicy things". They can wake up your taste buds. 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 / STONE / HEIGHT / TENSION
FREIGHT, STONE, HEIGHT, and TENSION 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 that ring
BELL / ALARM / CASHIER / PHONE
BELL, ALARM, CASHIER, and PHONE 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
Things that bloom
ALGAE / ROSE / FIREWORK / TULIP
ALGAE, ROSE, FIREWORK, and TULIP all fit the category "Things that bloom". Each can bloom in its own way. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.