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.
COVER, INDEX, CHAPTER, and SPINE all fit the category "Book parts". They help make up a printed book. 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
ENTER / SHIFT / SPACE / ESCAPE
ENTER, SHIFT, SPACE, and ESCAPE 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
Timekeepers
CLOCK / TIMER / HOURGLASS / WATCH
CLOCK, TIMER, HOURGLASS, and WATCH all fit the category "Timekeepers". They help measure time. 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.