I don't know anything about your document using the term "responsible person", and won't guess.
"Responsible person" is the ATF term uses for the person who is on the form to buy the tax stamp for an entity other than a person. It is everyone at the time of the application who has legal authority for the entity such as the LLC, partnership, trust, etc. It is the broad NFA category.
Responsible Person. In the case of a legal entity, including any trust, partnership, association, company (to include any Limited Liability Company (LLC)), corporation, or Licensed Entity that does not pay the Special (Occupational) Tax, any individual who possesses, directly or indirectly, the power or authority to direct the management and policies of the trust or entity to receive, possess, ship, transport, deliver, transfer or otherwise dispose of a firearm for, or on behalf of, the trust or legal entity.
The legal control of a Trust, by definition, is held by the Trustee, and the NFA trusts I have seen use Trustees/co-Trustees as the "individual who possesses" power/authority. My trust does make clear that only the Trustee is the responsible person so the ATF doesn't make beneficiaries get fingerprinted, etc.
The Single Shot Trust seems a little like a gimmick trust made cheap, so you have to buy one every time. It might be the way you want to go, but that's nothing I would have recommended to most my clients when I wrote trusts. I can see why some might use it.
As a lawyer, here is my disclaimer that nothing above is legal advice, it is for general educational purposes. If you want legal advice, you must get your own lawyer to give it.