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.
STONE, LIGHT, WALK, and BEAM all fit the category "Words after moon". Put one word before each answer. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Book parts
COVER / SPINE / INDEX / CHAPTER
COVER, SPINE, INDEX, and CHAPTER 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
Night sky
NEBULA / VENUS / ORION / METEOR
NEBULA, VENUS, ORION, and METEOR all fit the category "Night sky". Look up after sunset. 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
HAIL / DRIZZLE / FOG / SQUALL
HAIL, DRIZZLE, FOG, and SQUALL 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.