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.
MIST, FLEECE, VELVET, and PILLOW all fit the category "Soft things". They suggest a gentle texture. 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
PENCIL / STAPLER / LAMP / NOTEBOOK
PENCIL, STAPLER, LAMP, 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
Tiny units
PIXEL / BYTE / ATOM / SEED
PIXEL, BYTE, ATOM, and SEED all fit the category "Tiny units". Smallness connects them. 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 strings
PUPPET / HOODIE / KITE / VIOLIN
PUPPET, HOODIE, KITE, and VIOLIN all fit the category "Things with strings". A string is part of the object or how it works. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.