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.
MAP, LETTER, LAUNDRY, and ORIGAMI 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
Coffee shop orders
AMERICANO / MOCHA / CORTADO / LATTE
AMERICANO, MOCHA, CORTADO, and LATTE all fit the category "Coffee shop orders". You can hear these at a counter before 9 a.m. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Stage terms
SCENE / CURTAIN / PROP / ENCORE
SCENE, CURTAIN, PROP, and ENCORE all fit the category "Stage terms". A theater audience might notice 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
HOODIE / KITE / VIOLIN / PUPPET
HOODIE, KITE, VIOLIN, and PUPPET 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.