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.