"Psychologist" is often short for clinical psychologist. CLINICAL psychologists may counsel or treat patients and perform psychotherapy. In the United States clinical psychologists need to have professional licenses.

Having a good friend who is a life coach, let me explain it like this. First, calling yourself a "psychologist" requires a university degree in psychology, at least in the US. Most people who sell their services as psychologists have an advanced degree in psychology, in addition to their first degree. A psychologist often helps people with what might be called "problems". They help the client solve the problem and thus improve their life. A "life coach" has no educational requirements, but they may be psychologists as well. Most life coaches are older people with a lot of life or business experience. Rather than focusing on solving explicit problems, they focus more on helping people reach their potential. So you could say they focus on "opportunities" rather than "problems." The line between them is not a clear division. There is a lot of overlap in the people they help and the situations the people are in. Both can use the same evaluation methods and problem solving techniques, many of which are taken from research in psychology. This is only a rough explanation, but I hope it is somewhat helpful.

No, there is no connection. See https://en.wikipedia.org/wiki/Coaching#Life_coaching "Life coaching" is not a well defined term, anyone can claim to be a "life coach." "Life coaching draws upon a variety of tools and techniques from other disciplines such as sociology, psychology, Neuroscience,[14] positive adult development and career counseling with an aim towards helping people identify and achieve personal goals. Specialty life coaches may have degrees and may have studied psychological counseling, hypnosis, dream analysis, divination, marketing, Neuro Linguistic Programming, Affirmations, Auditing and other areas relevant to providing guidance, however life coaches are not necessarily therapists, consultants, sports instructors or health care providers and psychological intervention and business analysis may lie outside the scope of some coaches' work. Critics see life coaching as akin to psychotherapy but without restrictions, oversight, regulation, or established ethical policies."