Got it. I guess I was focusing to much on the whole, pay your fair share aspect.
It's not the easiest one and there's a lot of official interpretation that adds to it, but you need to read at 61.113 to get the basics.
Start with 113(a) which says the general rule that a private pilot can accept nothing in exchange for flying. Then it sets out a series of boxes of specific situations in which a pilot can get something. The problem is that you need to go outside the FAR itself to make some of those boxes meaningful (like knowing that a shared flight has to have a shared purpose) but there's no way to figure any what's in them unless you know the boxes exist.
And that they are separate. If you fit into the shared expense box, it doesn't have to be a business flight. If yo fit into the incidental to business box, it doesn't have to be sahred.
I think this is all confusing because the FAA is actually trying to be fair. It could have made a simple black and with rule. Written 61.113(a) and put in no exceptions.
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No person who holds a private pilot certificate may act as pilot in command of an aircraft that is carrying passengers or property for compensation or hire
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Simple. A couple of official interpretations of what "compensation" and "hire" are and you get a nice simple rule that says you can't get anything in exchange for flying. No reimbursement; no tax deductions for charity flights; no sharing expenses with buddies for the trip to the Super Bowl.
What the FAA actually did though, was to open up some things that were pretty obviously not part of being a charter or air taxi or airline. So what happens? You can see it even here - questions (sometimes just academic) about, "Well, if I do it
this way, can I fit it into the "sharing" exception and get around the reg?"
So the application of the rule tends to be case-by-case with the most generally useful rule being "if it quacks like a duck..."