"Better keep" vs "Better to keep."

Question

Which one is correct (when it comes at the beginning of the sentence)?

I searched on this website, and I found both instances:

Better keep a close eye on it to make sure it doesn't take over.

Better to keep your mouth shut and be thought a fool, than to speak and prove it.

Which version is correct? Or maybe both are correct?

Answer 1

1. Better keep a close eye on that.

better keep is a shortened form of: had better, the modal.

[You had] better keep a close on on that.

2. Better to [do whatever] than to do [whatever else].

better to is a shortened form of: It is better to [do this (rather) than that].