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.
PIANO, ADAGIO, FORTE, and ALLEGRO 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
Computer keys
ESCAPE / SHIFT / ENTER / SPACE
ESCAPE, SHIFT, ENTER, and SPACE all fit the category "Computer keys". They are labeled on many keyboards. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Green things
LIME / MOSS / PEA / FERN
LIME, MOSS, PEA, and FERN all fit the category "Green things". They commonly show up in green. 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 lanes
POOL / BOWLING / TRACK / ROAD
POOL, BOWLING, TRACK, and ROAD all fit the category "Things with lanes". Movement is organized into lanes. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.