instructor or lecturer or teacher?
Is it true the uk or us college teachers usually call themselves instructor when their official title is “lecturer” (not professor ). For example, Can I introduce myself with "I'm an instructor at XX university and I teach math" if i'm a college teacher?

Another question is someone once told me colleges instructors is often a general term for people who teach but aren't full time employees. Is that true?

Question

instructor or lecturer or teacher?
Is it true the uk or us college teachers usually call themselves instructor when their official title is “lecturer” (not professor ). For example, Can I introduce myself with "I'm an instructor at XX university and I teach math" if i'm a college teacher?

Another question is someone once told me colleges instructors is often a general term for people who teach but aren't full time employees. Is that true?

Answer

So, great question, generally in the UK:

Teacher is someone who teaches people at school up to about 18.
Lecturer is someone who teaches people at University so roughly 18-25
Instructor can be someone who is teaching a skill so welding/car driving/ swimming etc.  But they can also be say second degree students who are coaching first degree student in technical subjects in a laboratory.

But there will always be individuals who break the rules for their own reasons.

Professors have a "chair" which means they get specific funding and can focus on specific areas of research.  They may or may not be heads of department, but a head of department will be a Professor.

You also need to understand the difference between a college and a University in the UK and the USA, since I don't understand the US system I cannot comment.

In the UK the term instructor is not a general term for a part time employee on the teaching staff of a University but it does designate a junior position and may be part time.