There's a distinction between personal property and private property. My house, bed, toothbrush, and computer aren't private property, they're personal property.
No, you're just being obtuse for the sake of making an argument.
No, I'm making a point that is well-accepted in political philosophy.
Not even remotely. Political thinkers have been making that distinction since the 19th century. Go back to the 18th century and people generally wouldn't even have used the term "property" to refer to things other than land.