You will of course meet exceptions from this classification. In both cases you will have to write a thesis as part of the graduation requirements. The MPhil is a degree strongly oriented towards research, usually done as the first part of PhD studies. Once you have identified the subject and the kind of program you want to go to, you can start searching for universities. Second, it gives you a fair idea about how competitive those programs are.
Famous universities are rich, so more money is available for financial aid.
On the other hand, competition for admission and financing is tough. You will make the decision that suits you best, but we suggest you a few tips on how to make that choice. Choose more universities, it means more work with the application papers, but it increases your chances. We know cases when students handled up to 20 applications, but over 10 it usually gets tough. Your scores at the standardized tests should influence the final proportion of universities from each category.