Can an aircraft operator hire a Commercial Pilot to fly their aircraft under part 91? Here is the scenario. Joe owns a Cessna 172. Joe hires Tim to fly him in his 172 from A to B. Tim is a Commercial Pilot. Is this legal?

So far, everyone has said yes to this question, but not one person I have asked can cite a regulation or regulations for me.

Here is my chain of Logic:
Part 61.133(a)(1)(i):


  (i) Carrying persons or property for compensation or hire, provided
  the person is qualified in accordance with this part and with the
  applicable parts of this chapter that apply to the operation; and


Alright, let's see which part(s) apply to Tim.

Looking at 119.1(a)(1):


  (a) This part applies to each person operating or intending to operate
  civil aircraft - (1) As an air carrier or commercial operator, or
  both, in air commerce; or


Now let's see if any of those apply to us. Perhaps "commercial operator".
1.1:


  Commercial operator means a person who, for compensation or hire,
  engages in the carriage by aircraft in air commerce of persons or
  property, other than as an air carrier or foreign air carrier or under
  the authority of Part 375 of this title. Where it is doubtful that an
  operation is for “compensation or hire”, the test applied is whether
  the carriage by air is merely incidental to the person's other
  business or is, in itself, a major enterprise for profit.


Okay. This Tim seems to be operating as a Commercial Operator.

Now look at 119.23(b):


  (b) Each person who conducts noncommon carriage (except as provided in
  § 91.501(b) of this chapter) or private carriage operations for
  compensation or hire with airplanes having a passenger-seat
  configuration of less than 20 seats, excluding each crewmember seat,
  and a payload capacity of less than 6,000 pounds shall - (2) Conduct
  those operations in accordance with the requirements of part 135 of
  this chapter, except for those requirements applicable only to
  commuter operations; and


Well we want to be under part 91, so let's look at the exception provided as 91.501(b).

91.501(b):


  (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 - (4) Flights conducted by
  the operator of an airplane for his personal transportation, or the
  transportation of his guests when no charge, assessment, or fee is
  made for the transportation;


Number 4 seems to apply! Awesome!

But let's take a look at 91.501(a):


  (a) This subpart prescribes operating rules, in addition to those
  prescribed in other subparts of this part, governing the operation of
  large airplanes of U.S. registry, turbojet-powered multiengine civil
  airplanes of U.S. registry, and fractional ownership program aircraft
  of U.S. registry that are operating under subpart K of this part in
  operations not involving common carriage. The operating rules in this
  subpart do not apply to those aircraft when they are required to be
  operated under parts 121, 125, 129, 135, and 137 of this chapter.
  (Section 91.409 prescribes an inspection program for large and for
  turbine-powered (turbojet and turboprop) multiengine airplanes and
  turbine-powered rotorcraft of U.S. registry when they are operated
  under this part or part 129 or 137.)


What is a Large Aircraft.
1.1:


  Large aircraft means aircraft of more than 12,500 pounds, maximum
  certificated takeoff weight.


Okay, a Cessna 172 is not a Large Aircraft or turbojet multiengine. And this is not a Fractional Ownership Program since that requires 2 aircraft in the first place.

So, from my understand, Tim cannot fly for Joe since 91.501(a) says this part does not apply, BUT wait. 91.501(b) says it is allowed.

How is this intrepreted? Does 91.501(b) apply only to large aircraft and such on a fractional ownership program? What am I missing?

Everyone I know says it is allowed, but the rules seem to be different. Any help is appreciated.
To answer my own question. As pointed out by @MichaelHall and @pericynthion Tom would not fall under "Commercial operator".

Here is the definition for reference:


  Commercial operator means a person who, for compensation or hire,
  engages in the carriage by aircraft in air commerce of persons or
  property, other than as an air carrier or foreign air carrier or under
  the authority of Part 375 of this title. Where it is doubtful that an
  operation is for “compensation or hire”, the test applied is whether
  the carriage by air is merely incidental to the person's other
  business or is, in itself, a major enterprise for profit.


Another term which comes in useful here is:


  Operational control, with respect to a flight, means the exercise of
  authority over initiating, conducting or terminating a flight.


In this case, Joe would have operational control since it is his aircraft. Since Commercial Operation means "...person who... engages in the carriage by aircraft...", then Joe is the "person" and not the pilot Joe.

Another term within the definition is "air commerce". This means:


  Air commerce means interstate, overseas, or foreign air commerce or
  the transportation of mail by aircraft or any operation or navigation
  of aircraft within the limits of any Federal airway or any operation
  or navigation of aircraft which directly affects, or which may
  endanger safety in, interstate, overseas, or foreign air commerce.


Commerce is the activity of buying or selling. If we look at it from that level, the operator (Joe) is not engaging in air Commerce, because when you are operating an aircraft, you're perfectly allowed to pay for the usage of your own plane. So if this is not air commerce, then it falls under Part 91, and Tom is good to go.

However, if Joe were to engage in air commerce such as shipping stuff, but still hire Joe, then this falls under part 119 then 135 operations.

Refer to this legal interpretation: https://www.faa.gov/about/office_org/headquarters_offices/agc/practice_areas/regulations/interpretations/data/interps/1993/green%20-%20(1993)%20legal%20interpretation.pdf

Based off this interpretation, the aircraft and operator must be separate from the pilot. Also, the operator cannot conduct air commerce flights.