@bytebro No, it's subtle and I'm not sure the terminology is exactly right, but this subtlety is important to debunking the myths of capitalism being just.
"People's personal things" are the things that are like extensions of your being, that play a role in your everyday activities. Things where you would rightly feel personally violated if they were gone.
Private property is much larger in scope and extends to things you "own" as a matter of convention. A house you don't live in but rent. A factory you don't operate but keep the profits from. Symbols of wealth.
Machinery for taking care of the place where you live could fall somewhere in between, but when you don't even really own it (because the vendor who "sold" it to you still controls it) I think it's far more like *their* private property.