@Carly Good link, I never even heard about modal verbs until I got on italki 😅. How did you learn about all these?

The 2nd one. With have /has/had (perfect tenses) we use, generally, 3rd form of verb in a sentence unless you're using it as a verb. But here inthis case HAD BETTER IS A MODAL VERB SO WE USE THE 1ST FORM. If you're using it as an adverb or noun or an adjective, then the case would become different.

#2 & you can say “I’d better” or “I better” to make it more natural sounding

Yup, "had better" is another world of "should" and it is used more to express urgency or threat. https://www.youtube.com/watch?v=9r88YWp9XQo

The verb doesn't follow the had + past participle rule because it's not acting as a regular verb. It's called a modal verb. https://learningenglish.voanews.com/a/everyday-grammar-you-had-better-learn-modals/2865365.html