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.
SCALE, COMPASS, LEGEND, and BORDER all fit the category "Map features". These help you read a map. 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
KETTLEBELL / MAT / BENCH / ROPE
KETTLEBELL, MAT, BENCH, and ROPE 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
Words after moon
WALK / LIGHT / STONE / BEAM
WALK, LIGHT, STONE, and BEAM all fit the category "Words after moon". Put one word before each answer. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Desk items
STAPLER / PENCIL / NOTEBOOK / LAMP
STAPLER, PENCIL, NOTEBOOK, and LAMP 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.