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.
STONE, TENSION, HEIGHT, and FREIGHT 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 charge
BATTERY / CARD / LAWYER / RHINO
BATTERY, CARD, LAWYER, and RHINO 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 crowns
KING / TREE / WATCH / TOOTH
KING, TREE, WATCH, and TOOTH all fit the category "Things with crowns". A crown can sit on each of these. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Mystery story clues
EVIDENCE / ALIBI / MOTIVE / SUSPECT
EVIDENCE, ALIBI, MOTIVE, and SUSPECT all fit the category "Mystery story clues". A detective asks about these. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.