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.
BATTERY, LAWYER, CARD, and RHINO all fit the category "Things that charge". Charge has a different meaning for each. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Desert words
DUNE / MIRAGE / OASIS / CACTUS
DUNE, MIRAGE, OASIS, and CACTUS all fit the category "Desert words". Think sand and heat. 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 / PROP / CURTAIN / ENCORE
SCENE, PROP, CURTAIN, 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
Night sky
METEOR / ORION / VENUS / NEBULA
METEOR, ORION, VENUS, and NEBULA 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.