If you send me to get you lunch and I pay for it, say $8, with my own money, you would reimburse me the $8. If you gave me an extra $1 for my trouble, you have compensated me.

Isn't that a bit of an artificial distinction? If the time it took for you to get his lunch is worth $1 then he's still simply reimbursing you for your incurred costs. Just because you left the situation with more currency than you started with doesn't mean you've made a profit.

It would not be reimbursement if it were not structured that way.

See: fractional ownership

Honestly I have no idea how you'd structure a flight sharing business in a fractional ownership way such that nobody is actually being compensated for anything.

Ah, my impression was that the claim was that flight sharing wasn't a business (and therefore should not be treated as common carriage).