  These are all metamathematical examples, the kind of example that only a logician would construct.  It is natural to ask whether there are “natural” examples, roughly the sort of example occurring in the normal course of mathematics.  In the set theoretic case, such examples are abundant for the first two cases.  For example, PM is an example of the first kind of independence and CH is an example of the second kind of independence.  There are no known “natural” examples of the third kind of independence.  In the arithmetical case, such examples are rare.  There are examples of the first kind of independence (the most famous of which is a classic example due to Paris and Harrington) but none of the second or third kind of independence.