As a person who has worked in a professional on-campus tutoring environment with many professionally trained and CRLA Certified (levels 1 through 3) tutors, I must object to the loose definitions of tutor being used for American (and Canadian) tutors. I cannot however speak to the definition/role of a tutor in other anglophone countries.
First, a tutor is not a teacher/instructor at all. In fact one of the key guidelines of tutoring is to avoid presenting material that the student has not seen before. The role of tutor is to help the student to help themselves become better students, with the ultimate goal of eliminating the need for a tutor. Tutors respond to learner questions and needs, offering strategies and helping the student to find the answers for themselves.
Second, a tutor is not a mentor or coach. These roles are far more involved that tutoring and often go beyond the academic arena.
Third, a tutor can work one-on-one but they also can work with small groups of students. Some tutors also lead groups in what is called "Supplemental Instruction" (what you might call "satellite classes") but even CRLA acknowledges that SI is not tutoring.
There are many other aspects of being a good tutor but these 3 points are sufficient I think to help correctly define what a properly trained tutor does.
As for the use of "Teaching Assistant" per the original Question, I think that is fine for the college/university environment. However, be aware that the term became somewhat corrupted when it was borrowed by some of our basic education (elementary, middle and high schools) institutions. The role of a "TA" in those schools is more accurately called an "Instructional Aide" (IA) which is also known as a "paraprofessional" and the role often involves much more interaction with the children as well as assisting the teacher. Some IA's are even assigned to work one-on-one with children having special needs.