Dear Krish, you would have seen all the previous advice, so let me start from there.
In many companies like banks, once you sign the offer letter and report for the job, they send the joining report to HR. Only then does the original offer letter or appointment letter come to you in such cases. The joining kit may take some more time, but as per the joining report, the backend operations of HR and payroll will initiate their work to link you with the system, payroll, e-muster, and more importantly, they will start counting your salary days from the date of reporting, considering it as your joining date.
Common Practices for Resignation
Now, as per common practices, the Branch Manager has to issue letters of caution, warning, and escalate to HR for termination. Alternatively, you can quit through the system (via the intranet after logging in with your employee code and password). Resignation letters are not required for many new generation banks and insurance companies. However, they insist on quitting electronically, as explained above. If you want a relieving letter, you will have to quit; otherwise, they may issue a termination letter. If it is only for less than a week and no joining docs are submitted, HR may revoke your offer letter as if you have not joined and will make corrections in the system. In that case, you do not need to show this experience, but I would not advise you to do so because it could become an integrity issue in the future. There are enough chances for your new employer to find out about this if you are working in the same city.
Importance of Proper Exit
They are unlikely to want you to continue there thinking that you are good because absconding or not attending the job without a proper reason will be viewed as an attitude issue. They simply want you to go there and make a proper exit so that they can settle it smoothly. Auditing also happens in HR, and these kinds of issues may be serious for them. Therefore, you should seriously consider going to the branch and quitting officially. Many things happen for a good reason. Be serious and optimistic in life.
Regards.