What is the exact definition of "beginner", "intermediate", "advanced" and "pro"?

A beginner is everybody who is not an independent upwind kiter - which means that he’ll need some level of support on most spots (or even classes).

 An intermediate is everybody who is savely able to ride upwind in normal wind and water conditions. Intermediates are more or less independent kiters - but need at least a rescue option on most spots. Furthermore, intermediates may still be interested in taking classes, e.g. to learn jumps or wavekiting. 

An advanced is a truly independent kiter who is secure even when conditions get tough. Advanced kiters do not necessarily need a rescue service and may kite on remote spots without infrastructure or with demanding conditions. Advanced kiter are usually no more interested in taking classes - just maybe sometimes coaching with pro riders in order to tweak their style. 

The difference between "advanced" and "pro" is not necessarily a difference in skills, but "pro" means, that you kite as a profession - e.g. as kite instructor or team rider.