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.
WALK, STONE, BEAM, and LIGHT all fit the category "Words after moon". Put one word before 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 that bloom
ROSE / ALGAE / FIREWORK / TULIP
ROSE, ALGAE, 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.
Category
Things that charge
LAWYER / CARD / BATTERY / RHINO
LAWYER, CARD, BATTERY, 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
Ocean words
REEF / TIDE / KELP / WAVE
REEF, TIDE, KELP, and WAVE all fit the category "Ocean words". They belong near salt water. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.