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.
FORTE, ALLEGRO, PIANO, and ADAGIO all fit the category "Musical directions". A score might tell a player these. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Winter words
PARKA / ICICLE / SLED / FROST
PARKA, ICICLE, SLED, and FROST all fit the category "Winter words". Cold weather brings these out. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Garden tools
TROWEL / HOSE / SPADE / RAKE
TROWEL, HOSE, 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
Weather events
SQUALL / HAIL / DRIZZLE / FOG
SQUALL, HAIL, DRIZZLE, and FOG 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.