Tenant law regarding leases that started mid month

Hello everyone, I have been a tenant at my house since Oct last year and the contract stated as per my request that it is for 3 months and then becomes unlimited with a one month notice period in case I want to terminate it.

I started my rent on the 13th of Oct but when i rented it my first payment in Oct was the full months rent. I was desperate for a place to stay at the time so I didnt really ask.

come next month and the landord asked me to pay for Nov (at the time it was the 5th) so I informed him that since i payed on the 15th last month I thought i would do the same each month to which he informed me that no, i must pay by the 3rd of each month and for the 13 days that I have paid for in the first month he will give me an extra 13 days after termination to which i replied that we will see when the time comes.

Now my question is, I am moving out on the 15th of April and have given my notice a few days ago but the landlord refused to allow me to pay half rent for my last month and instead told me that i can stay an extra 13 days which not something i want. what can i do? am i in the wrong here?

TL;DR: started my rent on the 13th of oct but paid for the full month, now i am leaving and gave my one month notice but my landord says he can only give me 13 extra days after my notice but i want to pay half a month instead, is it within my rights to do so?