OP’s framing sets up a false dichotomy. “Moral facts exist objectively, independently from anyone’s personal beliefs” and “Moral facts and values are all completely relative and dependent on one's culture” are not two opposites that exhaust the range of possibilities.
For example, in my view morality is not objective in the sense that it exists independent of human beings. Only humans (and maybe some higher mammals) have morals. Morality is a behavioral trait: we do certain things, refrain from other things, and react when others do or don’t do things. Absent man in the universe, morality has no meaning, no purpose, and no source.
But that does not mean that morality is “completely relative and dependent on one’s culture.” Many moral concepts are universal among human cultures. That’s not surprising because, as a behavioral trait, morality has been subject to evolutionary pressure. As a result, moral precepts most important to group survival show up pretty much everywhere, while others that are less critical vary between groups. It’s the same for other evolved behaviors, such as religion.