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.
CASHIER, PHONE, BELL, and ALARM 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
Timekeepers
TIMER / CLOCK / HOURGLASS / WATCH
TIMER, CLOCK, HOURGLASS, and WATCH all fit the category "Timekeepers". They help measure time. 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 / ROPE / KETTLEBELL / MAT
BENCH, ROPE, KETTLEBELL, and MAT 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.
Category
River words
RAPIDS / BANK / CURRENT / DELTA
RAPIDS, BANK, CURRENT, and DELTA 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.