Your understanding of the distinction between personal and private property is simply flawed. Personal property is something that is owned by a single individual for their personal use, i.e. buying a house for you and your family to live in, while private property is something owned by either an individual or a corporation of people for the sole purpose of making money, i.e. buying a house that you never plan to live in for the sole purpose of renting it out to make more money.

A corporation can own consumer goods and it be private property, like Walmarts warehouses of consumer goods, and an individual can own producer goods as their own personal property, like tools and raw materials in their workshop.

The distinction is important because it boils down to intent and the inevitable consequences of that intent. Private property solely exists to make money, and therefore will put that goal above everything, leading to things like market manipulation, exploitation, and enforced scarcity. Trade is not capitalism, and capitalism isn't "when people trade".