Refer to your tenancy agreement. It should be in the form shown here:
https://www.fairtrading.nsw.gov.au/pdfs/tenants_and_home_owners/residential_tenancy_agreement.pdf
That form makes provision for a start and end date. Of course 6 months does not exactly match up with a specific number of fortnights, and it varies according to when you took out the lease, and whether there was an extra day in February for a leap year.
From what you say the agent is wrong, they can’t charge you more than specified in the agreement, nor charge you twice for the same day. It’s a bit disconcerting that they can’t get these basics right. Ask to see the principal of the agency, if he/she doesn’t see reason then you could take the matter further for resolution.
David