与房客的小额法庭纠纷

I live in Southern California. I had a month-to-month lease with my tenant. According to the lease, he needs to provide 60-day written notice before moving out. He moved out with 15-day notice. It took me 50 ds to find the next tenant. I therefore did not return him the 1-month-rent worth safety deposit. I also sued him on the small claim court after futile private negotiation asking for the remained rent and cleaning/repair fee. I failed the lawsuit as the court said "California Civil Codes 1946 [Notice required to terminate tenancy] states, in part: [A]s to tenancies from month to month either of the parties may terminate the same by giving at least 30 days' written notice thereof at any time and the rent shall be due and payable to and including the date of termination".

However when I looked up online (http://www.dca.ca.gov/publications/landlordbook/moving-out.shtml), I also found that "If your rental agreement specifies a different amount of notice (for example 10 days), you must give the landlord written notice as required by the agreement."

Can I request the court to reevaluate the case? Also, I am not sure whether my tenant is going to sue me as he threatened before. What shall I do? Many thanks in advance for your help.