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.
PENCIL, LAMP, STAPLER, and NOTEBOOK all fit the category "Desk items". They belong within arm's reach at work. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Musical directions
PIANO / ADAGIO / ALLEGRO / FORTE
PIANO, ADAGIO, ALLEGRO, and FORTE 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
Ocean words
REEF / KELP / TIDE / WAVE
REEF, KELP, TIDE, 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.
Category
Weather events
HAIL / DRIZZLE / SQUALL / FOG
HAIL, DRIZZLE, SQUALL, 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.