It would have been good if you had put this detailed post at the start. Anyway, it seems that the company is very clear they don't want you. Probably your work has been poor or not up to the mark, and someone in HR has been very smart. It is also clear that the company management (at least the immediately operational team) is aware of the same and in favor of removing you immediately and without pay. Sorry to say that such an attitude would be there in a large BPO only if your work or attitude was bad or you have done something seriously wrong (at least that has been my experience).
Now, if we analyze the series of activities, we will know where you stand. HR asked you to leave (don't come from tomorrow). They didn't give anything in writing. Unfortunately, you didn't insist on a written instruction, and you didn't bother to actually come the next day. You approached your manager, who didn’t respond (he is probably the one who insisted you be removed). You threatened to escalate but didn’t. So there is no proof that HR asked you to leave. All your discussion and protest was oral. On the other hand, they have proof that you did not come for work and didn’t give any reason or take approval for not coming. That was a serious mistake on your part, and a very smart move by HR.
Since you "absconded," they don’t need to terminate you, they don’t need to hold a domestic inquiry, they don’t need to do anything to show you needed to be removed. You left, and so they are now terminating you for not coming to work and not taking approval for absence. They don’t need to send you a notice if you are not coming to work voluntarily. Actually, they need to send you a notice asking you to return to work in xyz days, but I am sure they will create that letter when needed (backdated, of course).
You say HR admitted during the conversation that he asked you to leave. But do you have any proof? If not, then it has no legal standing. They paid you your previous month's salary on 12th August. That does not prove anything. They will claim that you worked and you were there the previous week, so they didn’t know you were absconding. So they paid because the salary was due to you. It does not prove that you were asked to leave or that your absence was approved.
You were asked to leave on 5th Aug. So if they consider it as absconding, they are entitled to deduct notice pay (15 days, as you have mentioned), which is more than the number of days worked. So they are then justified in paying you nothing. Their offer of paying you a salary for days worked (5 days) can also be construed as avoiding future disputes by keeping the records clear. Again, the offer to pay for resignation is oral, and no evidence is available to you.
You can always decide to fight, in which case, get a good lawyer. But remember that the courts will take years to give a decision. The company has money for a fight and lawyer fees. Do you have the same? And what happens to background checks from future employers? In case you want to move ahead with your life, get a clean settlement by resigning and getting the relieving letter. The salary of 5 days will make no difference in any case, I suspect. It's a decision that you need to make, keeping in mind the future career.