Regrettably this 'can I log it' debate is clouded by the CAA, JAA, the desperation of 'hours builders' and meaningless nonsense like 'SP1C U/T' or whatever it's termed.
Commonsense would indicate that if you're being taught something (such as a cross-Channel check) by an instructor, then you are the student and the FI is the Commander. Simply and easy - FI is P1C you are P U/T (but really ought to be P1/S if there was any sense in the world!).
But since when did the expressions 'CAA' or 'JAA' appear in the same sentence as 'commonsense'....