"We have three types of shares: round, square, and triangle. I suggested these names rather than number/letter designations because I wanted to avoid any connotation that one type of share is inherently better than another."
That reasoning for the names is brilliant!
Most of the bands I know use a much simpler method of spliting everything between bandmember payment and band money.
If one member has the van that hauls the gear, their gas expenses are covered by band money,
If one member pays for the album printing or merch, the bandmoney goes to them untill they are paid back.
Equiptment like strings and amps are personal - the members own their gear, they pay for it, but PAs and mixers are band gear, and it gets paid for by the pooled money.