Where did you get that idea? Anarchism is explicitly a left-wing political ideology that emerged from general socialist thought. The two are intimately linked in their development, and heavily influenced one another with the “purest” form of either by their own principles independently culminating in anarcho-communism. You can debate the viability of this system all you like, but the definition of the term is what it is.
It is defined as implicitly free of hierarchies, including the state. If you want to talk about a system with a state, you’re no longer taking about communism. We can talk about pragmatic incarnations of socialist policy, we can talk about the conditions necessary to foster a communist society, we can talk about the consequences of either. But if the subject is the definition of communism, none of that is relevant.
If we go by your distinction between personal and private property, what is to stop anyone from renting out their personal property?
A lack of money for one. The existence of other cool projectors, if you didn’t build the cool projector by yourself, that can be communally held. If you built it yourself, and decide to hoarde it yourself, presumably other members of the community would hesitate to share their cool stuff with you. Patents and IP are private property, so anyone with skill in projector-making can try to copy it.
If you recognize the benefit of sharing your cool stuff in exchange for others sharing their cool stuff with you, everyone gets to use lots of cool stuff. If you hoarde the cool stuff you personally invented, no one will let you use the cool stuff they personally invented, and you’ll only get to use the cool things you personally invented.