'I have been swimming' is the Present Perfect Continuous. The 'have been' part tells you that it's a present perfect form, and the 'swimming' tells you that it's a continuous form. 'I have swum' is Present Perfect, sometimes also called Present Perfect Simple to distinguish it from the continuous form. And yes, you can certainly say 'I have swum many times'. And by the way, it doesn't matter whether your textbook is British or American - the basic grammar of the English language is the same wherever you are.