In the USA they have clarified this position and there are times when you can (I am reliably informed) BOTH claim the P1 hours
Yes that is true (although I stand to be corrected), once you have a SEL rating for example and a PPL and you wanted to convert to another aircraft for example C152 to something bigger you could log that as straight P1 even though the instuctor is giving you instruction. The instructor would also log it as P1, which in my opinion is a double standard. Although under the US way you gain more hours I think the JAA way of logging is more accurate, the only confusion I have come across is whether a flight was PU/T or P1s.