Alba, I think the person who told you that was misinformed. When a lease (or other contract) is signed, that is the date of execution, upon which the agreement becomes binding. The commencement date is when the rights afforded by the contract actually begin, and it is often a different date.
So the "lease signed" date (date of execution) and the "lease start" date (commencement date) are usually different. It is important to use the correct one when completing documents for court though.