CFI in US terms means you can teach from zero to hero ie the student has zero training and you can teach them up to get a license and class rating in a single engine piston. In UK its termed chief flying instructor which doesn't really have much meaning apart from a job title. FI and FI(R) are used to denote flying instructor and fliying instructor (restricted) we also have CRI which is class rating instructor which allows training of licensed pilots for classes of aircraft such as SEP or MEP.
A TRI converts already licensed and trained pilots onto a particular type of aircraft that requires a type rating.