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.
LETTER, ORIGAMI, LAUNDRY, and MAP all fit the category "Things you fold". A crease is part of the process. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Mystery story clues
ALIBI / MOTIVE / EVIDENCE / SUSPECT
ALIBI, MOTIVE, EVIDENCE, and SUSPECT all fit the category "Mystery story clues". A detective asks 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 in a gym
BENCH / ROPE / KETTLEBELL / MAT
BENCH, ROPE, KETTLEBELL, and MAT all fit the category "Things in a gym". Training rooms are their habitat. 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
FORTE / ALLEGRO / PIANO / ADAGIO
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.