In resignation letter, should we to our direct supervisor (i.e team lead) or directly to the HOD (i.e. direct supervisor's boss)? Anyone can share experience?

Personal experiece: Sign offer letter and inform on the joining date then tender resignation letter. After that inform your current co there is 1 mth buy out plan and see how they respond. If no concern then you may proceed, if they insist then you can still buyout but might leave in bad terms depend on how you negotiate (they cant force you to stay anyway).

But if they insist and then you finally decided to fully serve the notice period, then you may go back to your future employer and tell them you want to complete all your project first before resigning and if they are ok to wait.

Ultimately is up to you, you need to stay firm on your decision and dont easily moved as one side will insist you to fully serve and the onther side will ask you to buyout.

For second you can email your HR for black and white ask them if any buyout plan and they will reply from there. My 2cents