In fine Irish fashion, let me answer your second question first -
The necessary "qualifying beneficiary" relationship is established when the beneficiary is either the spouse, child, grandchild, sibling or parent of the account owner. The status of being the "trustee" of the trust has no bearing on the determination if someone is a qualifying beneficiary.
Now to your first question:
For POD and Totten Trust accounts, the names of the beneficiaries must be identified by name in the bank's deposit account records. 12 CFR Section 330.10(b) provides:
"(b) Required intention. The required intention in paragraph (a) of this section that upon the owner's death the funds shall belong to one or more qualifying beneficiaries must be manifested in the title of the account using commonly accepted terms such as, but not limited to, 'in trust for,' 'as trustee for,' 'payable on death to,' or any acronym therefor. In addition, the beneficiaries must be specifically named in the deposit account records of the insured depository institution. The settlor of a revocable trust account shall be presumed to own the funds deposited into the account."
The rule is the opposite for formal revocable trusts with written trust agreements - there is no requirement that the beneficiaries of a formal trust be named in the bank's account records. See 12 CFR Section 330.10(f)(4):
"(4) In order for a depositor to qualify for the living trust account coverage provided under this paragraph (f), the title of the account must reflect that the funds in the account are held pursuant to a formal revocable trust. There is no requirement, however, that the deposit accounts records of the depository institution indicate the names of the beneficiaries of the living trust and their ownership interests in the trust."