That's not what they mean by "private property." Leftists are just so goddamn stubborn that they refuse to acknowledge the immediate confusion and use any other label.
Is there anything resembling a consensus on that? They're not interested in our toothbrushes as they say, but how about arable land? A machine shop? A work truck?
When they say "private property" they are not referring to the people's toothbrush, but the private ownership of production. Houses and such are "personal property".
That depends on the use. Are you using it as a productive business for profit or just something you like, maybe even for some side money. Are you hiring employees to work it for you? Once you make it a mode of production it's considered a privately owned means of production. If it's something you just have, it's personal property.
That's the gist anyway. These are actual, defined terms and not just things people throw around on the internet. If you want something more specific, doing a search will probably get a better explanation than I'm good at giving.