What is a "personal tutor"?

In the UK, a personal tutor is assigned to you when you start your degree and the same person will generally maintain that role for the whole of your course. They are usually a lecturer or more senior-- never a student, PhD or otherwise. There are often a mandated number of meetings that a student must have with their tutor: once a term is common.

The role of a personal tutor is mostly non-academic. While you could discuss academic problems with them, they are really there to help with pastoral matters, such as wellbeing, or helping you to decide what courses to take or how best to manage your time. Personal tutors are a common choice of reference letter writers for their students' applications for postgraduate degrees, as they likely know the student and their grades very well.

"Personal tutor" in the sense of the question you linked to is definitely not a US-based term.

We absolutely have tutors. A tutor would be used when you need extra help to keep up with the content in the class. Getting a C in chemistry? Then you might hire a tutor to spend extra time with you outside of class to learn the material.

Similar to the "personal tutor" from that link, undergraduates are also typically assigned general advisers that they meet with a few times a year. They help to make sure you are fulfilling your academic requirements (have the right number of credits to graduate, are successful in courses) and can connect you to resources for academic problems, and also if you are encountering things that might violate university or legal regulations (like the bullying example you linked to). At my private R1 university, these advisers are always staff and never faculty or students.