The answer to your question is yes, you really should report the foreign account(s), in India.  As long as the account(s) itself is over the dollar limit specified in the law, the reporting rules apply equally to US citizens, permanent residents (as well as certain nonresident aliens) who have an ownership interest in the account.

If for no other reason that to be on the safe side of caution, it would be best practice to disclose the account(s) to the US government.  Also, it doesn't cost anything to comply with the law, other than your time spent filling out the forms; and at least there is no filing fee charged.

That said, as you note in your question, there are in fact two separate disclosure forms that may be required; each also has different reporting rules.  One is known as IRS Form 8938, and can be attached to the relevant yearly Form 1040 tax return.  The other is FinCen Form 114, which can only be filed via the internet.  The following Internal Revenue Service webpage describes them in some detail, and provides their dollar value reporting levels:

https://www.irs.gov/businesses/comparison-of-form-8938-and-fbar-requirements

Form 8938 is included in TurboTax; FinCen Form 114 is not, and would need to access that reporting webpage separately.  Note, however, that you can get to the FinCen reporting internet site directly through the above IRS link.

Thank you for asking this important question.