If you thought that you were commissioned whilst a Naval Cadet because of that letter, then I fear that you have long been under a false assumption. A Cadet is an officer under training, as is a Midshipman, and only at the end of the training is a commission granted by the Lord High Admiral, Queen Elizabeth II. My commission was granted as a Sub Lieutenant and I wager that yours was, too.
I understand that in more recent years the grey area of Midshipman and commissions has been changed and they now receive a commission, but going back in time it was a warrant rank and then became 'undetermined'. At the time of the photo that I was referring to, and my time in the rank, Midshipman was NOT a commissioned officer but an officer under training.