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.
VIOLIN, PUPPET, KITE, and HOODIE 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.
Category
Weather events
SQUALL / DRIZZLE / HAIL / FOG
SQUALL, DRIZZLE, HAIL, and FOG all fit the category "Weather events". Forecasts warn you 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
Timekeepers
WATCH / HOURGLASS / TIMER / CLOCK
WATCH, HOURGLASS, TIMER, and CLOCK 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 you fold
MAP / LETTER / ORIGAMI / LAUNDRY
MAP, LETTER, ORIGAMI, and LAUNDRY 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.