Agreed, it is an odd subject for a travel bulletin board. And very complicated. Renunciation is a pain in the butt for everyone involved. TWA is correct that there must be a clear intent by a person to give up their citizenship, but the intent does not have to be explicitly stated. There are many acts that create a presumption of intent to give up citizenship, including becoming the head of a foreign state or volunteering for the armed forces of a foreign state engaged in hostilities with the US. The State Dept is authorized to begin a renunciation case ex parte against such a person, but it is still considered voluntary to comply with Afroyim. At the same time, the US does not have to allow you to renounce your citizenship. A renunciation is not final until the certificate of loss of nationality is approved by consular affairs in DC, and they can deny your request. To complicate things further, there are many states (India included) that do not allow dual citizenship, but the US does. If it seems clear as mud, it's because it is.

All that anyone should ever want to know about renunciation can be found at 7 FAM 1200 and on, which is chapter 7 of the Foreign Affairs Manual.