Issue in Notice Period Duration

Hi Sir / Madam I had recently resigned from my organization after 1.9 years. My DOJ was on 15 Oct 2018. At the time of joining, I was given an offer letter which had company T&C post joining no other documents were provided to me nor I had signed anything with the organization. Recently I got another offer and I decided to move on with that and put my papers on 30th June stating that I am applicable for 1 month notice period as my offer letter clause says that "On successful completion of your probation period based on your performance evaluation, you will be offered a regular position with the company.” I am requesting for 1 month notice period considering that my probation is still going on as I had never been offered any permanent position with the current company and my termination clause says that "In case of termination of employment at will the employee shall be free to terminate this employment agreement at will and at any time with or without cause upon thirty (30) days prior notice during probation period and ninety (90) days prior written notice after confirmation of services, by the employee or payment of equivalent salary in lieu thereof, to the company. This is subject to company approval" I just wanted to know whether I am right on my part or not? Should I go legal ways based on this as company is not agreeing for 1 month notice period and is harassing me for 3 months notice period?