You are correct.
Your sister's nonqualified annuity is not an inherited annuity if she simply purchased her own annuity with cash. She should have a code 4D Form 1099-R that reports the cashing in of the inherited annuity, but any distributions from the annuity that she subsequently purchased should be reported with code 7D.
As for what is supposed to be an inherited IRA, I suspect that the custodian that held your aunt's IRA initiated a nonreportable trustee-to-trustee transfer of your sister's share to an inherited IRA for the benefit of your sister, but somehow the receiving account got mistakenly coded in the custodian's records as an owned rather than an inherited IRA. That custodian will need to correct their records and issue a corrected Form 1099-R. However, if a distribution was paid to your sister form your aunt's IRA (which is distribution that is required to be reported on a code-4 Form 1099-R) and she subsequently deposited the funds into a new IRA, that IRA can't be an inherited IRA and the deposit of those funds constitutes an ordinary IRA contribution.
