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.
SURF, DASH, KEY, and SCORE all fit the category "Words before board". Put one word after 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
Tiny units
ATOM / SEED / PIXEL / BYTE
ATOM, SEED, PIXEL, and BYTE all fit the category "Tiny units". Smallness connects them. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Kitchen verbs
SEAR / WHISK / FOLD / DICE
SEAR, WHISK, FOLD, and DICE all fit the category "Kitchen verbs". These are actions from a recipe. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Words after moon
BEAM / LIGHT / STONE / WALK
BEAM, LIGHT, STONE, and WALK 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.