Again and again, I don't understand why people don't give the full details when they initially make a post.
Originally, I was inclined to believe that the agreement is notionally extended when you continued working. But the agreement very clearly states that if no extension is given, the contract is terminated.
However, since you continued to work and the company continued to pay you, it would mean (in the eyes of the law) that a new contract of employment is entered into (contracts need not be in writing, it can be oral or by action). Therefore, you are a normal employee and need to follow the norms. Thus, the standard notice period will apply.
Alternatively, some will say the new agreement would be in line with the previous one and in that case also, you need to give notice as specified. Incidentally, the contract says you need to give 15 days' salary, not 1 month's salary.
A third approach may be to consider that the new contract is not in line with normal employment but employment at will, and therefore, you can leave at any time without notice.
Which is the correct approach is a discussion between lawyers before a judge. The question is now: what can you do if you do not want to get into a long drawn legal battle whose outcome you are unsure of? You need to sit down with the management. Show them the contract that specifically says that if the contract is not explicitly extended, it is terminated and that you no longer need to follow its terms. The rest depends on your negotiation skills; we cannot comment or advise on it.
It may be in your interest to pay the 15 days' notice pay as specified in the contract and get your relieving letter. Remember that in the future, you may join a company that will ask for Background Verification, and in that case, they will come back to your current employer. You do not want them to give a bad review, do you?