Your friend would be best served by trying to obtain information directly. She can also review the California tenant's guide.
From what you have written, it isn't at all clear that they aren't both offering the same thing. One says "pay part of next month's rent", the other offers ten days, which is part of the next month's rent.Quoting prince7122002
Leaving items in a rental unit at the conclusion of a tenancy does not necessarily extend the tenancy. The California Department of Consumer Affairs provides some information on a landlord's remedies when a tenant leaves property behind.Quoting prince7122002
Was the twenty day notice ever accepted as sufficient?Quoting prince7122002
If both roommates were on a written lease with a landlord, rent is not owed to the other roomate; it would be owed to the landlord. Further, the landlord would not give somebody "permission to take their name off of the contract" - the landlord would release somebody from the lease.Quoting prince7122002
You can review the handbook linked above in relation to the return of a security deposit.Quoting prince7122002

