If you want me to do a "check out" of any type than I, for the purpose of responsibility for the safe conduct of the flight, am PIC. The deal is if at any time I say "I have control" you take immediately take your hands and feet of the controls and be silent.
If you are not happy with this than I have no problem with that, however I will not fly with you and will suggest some one else who you might ask.
The aircraft log book will reflect the fact that I was PIC. I frankly don't care what you put in your log book.
As Genghis said it is absolutely vital that there is a clear delineation of responsibilities before you go flying