Actually 135 can apply to operations that do not hold out. That would be the non common carriage, which is carriage for hire that doesn't hold out to the public. AD 120-12A is more than a bit outdated.
I can see both sides - clearly a pilot can be hired to fly an airplane in a non commercial operation, but if a pilot is being hired for the purpose of flying people around, why is that not an on demand air taxi?
BTW, I haven't just been reading and making this up. Lots of legal opinion research.
Wagner: A contract between a commercial pilot and his company to move company goods constitutes non common carriage.
https://www.faa.gov/about/office_org/headquarters_offices/agc/practice_areas/regulations/interpretations/Data/interps/2010/Wagner - (2010) Legal Interpretation.pdf
Krontje: non common carriage can still call under 119.1. There is a good deal of twisty law involved in the line, but someone stated that holding out is required for 119.1 and that's not true.
https://www.faa.gov/about/office_org/headquarters_offices/agc/practice_areas/regulations/interpretations/Data/interps/2014/Kroontje - (2014) Legal Interpretation.pdf
Mills: Reminds us that private carriage is usually long term, not a on off event:
https://www.faa.gov/about/office_org/headquarters_offices/agc/practice_areas/regulations/interpretations/Data/interps/2011/Mills-DFWFSDO - (2011) Legal Interpretation.pdf
Howell-Patriots: Although phrased as a 61.113 question, the discussion wanders into 61.133 realm on the question of what constitutes compensation.
Cooling is a weird one - it basically states that an aircraft owned by a LLC club falls into 119.1 certification requirements because a LLC corporation is not the same as a person. Even though it's a "club" where members pay for operations, it's a commercial operation.
https://www.faa.gov/about/office_org/headquarters_offices/agc/practice_areas/regulations/interpretations/Data/interps/2017/Cooling - (2017) Legal Interpretation.pdf
Bonilla; reinforces that transporting a passenger for hire requires 135 certification. It also states that carriage can be common or noncommon for the exception tasks in 119.1(e).
https://www.faa.gov/about/office_org/headquarters_offices/agc/practice_areas/regulations/interpretations/Data/interps/2011/Bonilla - (2011) Legal Interpretation.pdf
Going back to the DPE - what I come back with is that because it's a one time request, it's an on demand taxi operation. A 135 cert is required by someone before the pilot can fly it. The consistent rule is that a flight for compensation or hire requires 135 certification.
"Duck" theory. If you weren't friends and a person asked you to fly them, would it quack?