Is a tenant's notification of their desire to leave the house considered a termination of the original lease contract if they reverse their decision and the landlord agrees to their stay, and are they then obligated to pay new rent and a new rent deposit?

If the tenant wishes to terminate the contract before its term, and the landlord agrees to the rescission, then the tenant changes his mind and wishes to complete the term, and the landlord agrees to that, then the rescission of the rescission is valid, and the original contract returns, so there is no need for a new contract.