The biggest mistake you made was lying to him. To me, that is the worst thing anyone can do, DOM or sub. Lying is not the way to build trust.

If I were your DOM in this situation I'd first explain your infraction to you.
Then tell you to come up with an appropriate punishment. When I was a kid my parents often used this approach. You better believe the fear of not thinking up a severe enough one for me was as bad as any punishment I could think of. If I was too easy on myself they would add to it many times over. They saw it as I really wasn't sorry for what I did. If they thought my idea of how I should be punished reflected me being truly sorry, they oven reduced it, and sometimes totally eliminated it. They took the opion I had learned my lesson. Disappointing a loved one is a punishment in itself.

However, your DOM has told you to do something not outside your limited. You should do it.