A dramatic public/private split is one of the most reported patterns in narcissistic dynamics: charming, generous, and warm to outsiders — cold, critical, or explosive behind closed doors. The public self protects their image; the private self doesn't have to. If the gap keeps making you doubt what happens at home, the gap itself is the signal.
Image management. Outsiders are an audience whose approval feeds self-esteem; family is 'safe' — there's no image to protect at home, so the mask relaxes. This is why friends and even relatives may see none of what you live with.
A private record. Write incidents down as they happen, with dates. When the public performance makes you doubt the private reality, the record doesn't waver.