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.
LATTE, CORTADO, AMERICANO, and MOCHA 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
Things you fold
MAP / ORIGAMI / LAUNDRY / LETTER
MAP, ORIGAMI, LAUNDRY, and LETTER 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
Photo terms
LENS / SHUTTER / FOCUS / FLASH
LENS, SHUTTER, FOCUS, and FLASH all fit the category "Photo terms". A camera owner knows these. 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 crowns
KING / TREE / TOOTH / WATCH
KING, TREE, TOOTH, and WATCH all fit the category "Things with crowns". A crown can sit on each of these. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.