Privately owned US companies that make massive profits off the capitalist system, like Stanford, obviously might not describe these things in an entirely honest way. The first definition of communism is almost correct, but the second is a common misinterpretation of communist theory.
The people who solidified communism as a political theory made a pretty clear distinction between private and personal property, as well as what they thought of landlords. Renting requires someone to own the means of production and lend it out to someone at a profit, which goes against the very foundations of communist theory.
Communism isn’t “the government takes all your stuff and divides it equally among people” either. I suggest reading some theory. Marxists.org is a good place to start www.marxists.org/subject/students/index.htm