No, the answer to this one isn't in LASORS - nor is it written down anywhere at all.
There are three different schools of thought for your first question:
1) Log the time as P1. The instructor does not log the time at all. This means that the instructor has no legal authority to affect the flight in any way - it must be clearly agreed between the two of you before the flight.
If you don't agree on this, then you can argue for one of the following two options:
2) Log the time as P1US. This sounds reasonable - you are qualified to and capable of flying the aircraft, but there is an instructor supervising you. LASORS only lists a very limited number of cases where P1US can be used, though, and this is not one of them.
3) Log the time as PUT. Technically, you are not being trained, but logging as PUT leaves no doubt at all that the instructor is in charge, and can override your decisions if he feels it's necessary.
Personally, I'd go for option 3. Although option 2 logically makes more sense, if you log time as P1US and either the CAA or your insurance company feel that this is not correct and it should have been PUT, then you could find yourself in trouble. On the other hand, if you log time as PUT but someone things it should have been P1US, then that's not going to cause you any problems. So option 3 is the safest one, even though option 2 is the most logical. But you will get 11 different opinions from 10 different people on this point.
On the second question, the answer is exactly the same as for the first question. You are legally allowed to fly the C172 without any further training - you are therefore able to log P1 as long as your instructor agrees, and if he doesn't agree to this then you have the same dilema re PUT or P1US. There is one exception to this, though, and that is if you require differences training for the aircraft (i.e. if it has retractable gear, variable pitch prop, tailwheel, turbo/super charger or pressurized cabin and you have not flown anything with this difference before). In this case, you are PUT and your instructor is P1.
The other considerations are the 90-day rule, and insurance company requirements. Either of these could prevent you from logging P1 or P1US, and force you to log PUT in either case.
FFF
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