Thanks for your enquiry.
On the basis that the fixed period of your Tenancy Agreement has expired, the Tenancy continues until either party serves Notice to terminate the agreement. Any Notice served by you as Tenant has to be served so that the Notice period ends on the 2nd day of the month.
As your Tenancy commenced on the 2nd of the month, if you now wish to end the agreement you do indeed need to give your Landlord Notice on or before the 2nd day of the month to expire the next month. Hence, I'm afraid your Landlord is correct, and you should serve him with another Notice at any time this month notifying your Landlord that you wish to terminate your Tenancy on 2nd October.
I hope this answers your question, and if so, I would be grateful if you could leave positive feedback.