Thank you Jim for such a detailed response and also for sharing the direct support number.
To be clear, in my case the charge did show as "Apple Services". However, the problem was something else.
Posting the resolution here, in case others face the same issue:
Turns out because of the UPI mode of payment, even though the subscription was cancelled on my phone, the standing instruction with my bank was not cancelled. Therefore, on the day of renewal, the bank went ahead and executed the standing instruction and auto-debited the money.
This is an India-specific payments issue because the Central bank of India does not support Credit Card or Debit Cards for App Store purchases anymore. They only accept UPI transactions or Apple Account balance. The UPI payment interface only allows scheduling a payment by creating a one-time mandate or a standing instruction.
The one-time mandate (OTM) is a process for you to authorize your bank to debit money from your account for recurring payments.
Due to this process, even if the subscription is cancelled from Apple, the bank still thinks that there is a payment scheduled and debits on the renewal date.
The solution is that after cancelling the subscription from Apple, you must to go your bank app and explicitly cancel the standing instruction too before the renewal date.
If you have successfully cancelled the subscription within the stipulated timeframe but the bank has deducted the amount, it will be refunded within 48 hours (the timeline varies from bank to bank). If not, after 48 hours you can contact the Apple support and they can either initiate the refund, or else activate your subscription.
Hope this helps.