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.
HOSE, TROWEL, SPADE, and RAKE all fit the category "Garden tools". These live in a shed. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Baking staples
FLOUR / SUGAR / YEAST / BUTTER
FLOUR, SUGAR, YEAST, and BUTTER all fit the category "Baking staples". A bakery keeps these close. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Weather events
FOG / HAIL / DRIZZLE / SQUALL
FOG, HAIL, DRIZZLE, and SQUALL all fit the category "Weather events". Forecasts warn you about these. 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
SOAP / MUSIC / CHOCOLATE / SIGNAL
SOAP, MUSIC, 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.