Unless your lease contains a provision requiring notice before the end of the lease term, and very few do, it ends automatically on June 30, notice or no notice. Although the landlord's notice may be considered an offer to have you stay for another 30 days, you do not have to do so. But you do not have any right to prorated rent: you must pay through the lease term.
You can ask the landlord if it will repay you any of the rent if you vacate early. It has no obligation to do so.
It is your discretion to stay beyond the June 30 lease expiration date to August 3. However, you can also move out by June 30 according to the lease term. There will be no pro-rated refund if you move out before June 30. Edward C. Ip www.lawyer4property.com
No attorney / client relationship established. The answr is for discussion and general information only. The lawyer had not reviewed any documents or contract prior to the above comments.
You won't get any refund and can stay through the time you've paid. Since you have a little time, you should certainly request a "final walk through" with the landlord (put the request in writing) where they have to go through the unit with you and advise of any potential charges, and let you try and fix them. Also, know that the landlord has an absolute requirement to send you any security deposit refund within 21-days of vacating. If they charge you for anything, they have to send you a statement within that same 21-days. Best of luck,