"Private carriage" doesn't need an operating certificate.
As per 119.1(d), "This part does not govern operations conducted under part 91, subpart K (when common carriage is not involved)...".
Part 91, Subpart K only applies to large and turbine aircraft and fractional ownership programs (which, while not necessarily an "operating certificate", there is FAA approval of Management Specifications).
And once you get to Subpart K, you'll find 91.501(b): "Operations that may be conducted under the rules in this subpart instead of those in parts 121, 129, 135, and 137 of this chapter when common carriage is not involved, include..." followed by a list of specific activities, one or two of which involve very tightly define private carriage activities and
none of which fall into the general category of a select set of customers you didn't get by holding out. You'll find that the other Parts §119.1(d) refers to also involve tight regulation.
Back to 119, take a look at 119.5 (e) and (h):
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119.5(e): A person
authorized to engage in noncommon or private carriage under part 125 or
part 135 of this chapter, or both, shall be issued only one certificate authorizing such carriage, regardless of the kind of operation or the class or size of aircraft to be operated.
==============================
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119.5(h) A person holding an
Operating Certificate authorizing noncommon or private carriage operations shall not conduct any operations in common carriage.
==============================
Don't confuse the "public vs private carriage" issue with the need for an operating certificate (I used to - this stuff ain't easy). If you look at the typical Part 135 Operating Certificate, the first space after identifying the holder will describe the type of operations. For a 135 private carriage operation, it will say "Private Carriage Operations."
Take a look at
http://fsims.faa.gov/wdocs/8900.1/v02 air op & agency cert/chapter 01/02_001_004.htm and scroll down to Figure 2 - you'll see a sample for a private carriage operating certificate.
Other than fractionals, a true Part 91 operation does not involve carriage or persons and property for hire at all, except for the specific activities excepted.
We all tend to confuse them because the differences are complicated and the lines fuzzy, but the issues surrounding these three things are all different:
- Private or Commercial Pilot: is the pilot being paid to fly?
- Private or Commercial Operator: Are people or cargo be carried for hire?
- Public or Private:Is there a holding out of availability to the general public?
AC 120-12A sure sounds like it's authorizing Part 91 private carriage, which doesn't help the confusion (Personally, I think it's outdated and needs to be replaced; take a look at the Part 91 references in it). Bottom line is that I am unaware of even one case in which the issue of private or public was raised by the FAA where the NTSB found that carriage was private.
ILS37R's comment about being a minefield is spot on.