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.
BATTERY, LAWYER, RHINO, and CARD all fit the category "Things that charge". Charge has a different meaning for each. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Things with bars
MUSIC / SOAP / CHOCOLATE / SIGNAL
MUSIC, SOAP, CHOCOLATE, and SIGNAL all fit the category "Things with bars". Bars can describe each answer. 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
Things that ring
CASHIER / BELL / PHONE / ALARM
CASHIER, BELL, PHONE, 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.