Depends on which anarchists you ask. Some anarchists will say any concept of property is a spook, while others may have a formalized definition. Either way the logical conclusion of both foundations of thought lead to the idea that it doesn't matter so much what exactly is and isn't private or personal property, but rather how such evaluations are made. Under any type of capitalism, due to the presence of hierarchies and a limited supply of resources means all property relations are decided by force of whoever has the most resources. They may not say that, saying that "NAP will prevail" but history shows that this has happened over and over again because their NAP doesn't work in a society based upon competition for resources. In anarchism, both individualist and collectivists generally agree that property relations are best decided by agreement.