The rule that I was taught as a kid was “I before E except after C only if the sound is ee”
It catches all of the examples on the mug except “weird” which gets a pass in my book because it’s weird (Keith’s excluded because proper nouns don’t follow rules). It doesn’t catch “caffeine” but the “ei” in “caffeinated” isn’t pronounced ee where I come from.
I’ve never heard the “ay” rule but it doesn’t seem to catch as many words as the “not ee” rule.