Just a few points to add....
I've never heard of SPIC....must be an American invention.
Don't get another log book, and don't throw away ANYTHING. You might need those signatures, and they're proof to the CAA if foreign flying is ever queried for the hours - I've never had it happen, but I've heard rumours of it. And extra signatures etc won't do any harm; at the very least, they're nice souvenirs of flying trips in years to come.
Basically, when solo you log PIC or P1, with your name in log book as in command. With an instructor, you log P UT, in the dual column, with his/her name as pilot in command. The only exceptions to this are a test or check ride which you pass, which is then PICUS or P1S; or if you decide in advance that the instructor is just along for the ride and is legally a passenger. Sometimes, in practice, I've asked afterwards and been told something like, "You can log it as P1 if you like; I didn't touch the controls and I don't need the hours". Is this legal? We could debate it for hours, and probably not, strictly speaking, so decide yourself how you feel about that and what you want to log, if it happens.
This is a basic question, but I agree - you will get many, many different answers. What I did - and I'm not saying this is right - was learn the usual rules, and if I got told anything different for grey areas, go along with whatever the instructor said. In the long run, it doesn't matter that much if the odd line is "wrong". And I'm told the CAA doesn't get upset about occasional anomalies, and they've seen my log book and not objected, and I have American signatures and stamps and also Russian ones and tippex and photos and God only knows what else in my log book!
If still in doubt, make a note in the remarks column.
Sorry this isn't as cut and dried as you'd probably like, but I hope it helps.