A REIT is a specific tax election, not a description of a business. A company elects to be treated as a real estate investment trust and in exchange for that treatment it has to keep meeting tests: most of its assets have to be real estate or real estate related, most of its income has to come from rents or mortgage interest, and it has to distribute the large majority of its taxable income to shareholders each year. The distribution requirement is the part people remember, and it's real, but on its own it doesn't make anything a REIT. A utility paying a 5 percent dividend is just a utility. Confirm the specific current tests with a tax professional if you ever need the exact thresholds, because they're statutory and they get details.
So on your list: a homebuilder sells houses, which is inventory rather than income-producing property, and it isn't a REIT. A company managing apartments for other owners is earning fee income, which is a service business. A fund holding a basket of REIT shares is a fund, usually an ETF or mutual fund, and it's a perfectly normal way to own REITs without picking one.
The loose usage you'll hit is people calling any real estate fund a REIT, including private, non-traded vehicles. Those can be genuine REITs by tax election while being nothing like a listed one, because you can't sell them on an exchange on a Tuesday afternoon. When someone in this room says public REIT, they mean shares that trade on a stock exchange. That liquidity is the whole point of the category.
