> Non-profit is an misleadingly named corporate structure that really means non-stock.

No, its not. Most states allow non-profit corporations, some allow non-profit non-corporations (associations, trusts, etc.) Some kinds of non-profits may have members in much the same way that an LLC does, however, a non-profit is not structured to return profits to those members (or anyone else, hence the name.)

There are a wide variety of rules for federal and state tax exemption and other treatment for various forms of non-profits, of which the the 501(c)(3) charity provisions making the entities own revenues tax exempt and contributions tax deductible to the contributor are the best known, but hardly the only significant ones.