Property implies a legal framework surrounding some object such that you are able to "own" it, without actually using it.

Possession exists without an obvious legal framework, as its ownership is obvious, and the principle benefit of this distinction is that Possession disallows for the accumulation of wealth.

Now, some people may not recognize your claim regarding possessive property just as you might not recognize a claim regarding sticky property. Now, if possessive property is under dispute, it becomes sticky. This is to say if, you are driven off the land you are using, you still claim it, no? If you do still claim that it was a violation of possessive rights, then possessive rights is the legal framework of your sticky property. This then means that all property is sticky, but some legal frameworks are built around a system of physical possession(your definition), and others not (sticky property, that most modern legal systems recognize).

Whether or not this is a violation of possessive rights is a non-issue. Of course, both sides would challenge the ownership claim of the other side. But, the fact remains, one side has the property, whether through violence or other means, and the other does not. If you claim property that you do not possess, it enters the realm of sticky property.

A commune that must defend itself against some foreign entity, is absolutely defending property, but that property is sticky.

No. A sticky property simply explains that an ownership claim "sticks" to the current claimant, regardless of his occupancy or use of the item. A commune defending some land or capital it is currently occupying or using, is not claiming sticky property but rather a possessive claim. The owner is still not defined by violence for any external party. Anyone violently forcing the commune out of the possessive claims would rightly have their ownership claim challenged as it was based on a violation of possessive rights.

If a claim is made against your ability to possess a toothbrush, it is whoever has the power that determines who is the owner. This is to say that this becomes sticky property the instant it becomes challenged.

Not really. A possession simply signifies a claim of ownership that is grounded on occupancy or use. What comprises occupancy or use and the time limits that are set before something is considered to be abandoned are set by social norms in society. For example, nobody in their right mind would consider a personal possession like a toothbrush to have been abandoned if it's in your house and not currently being used.

As such, this argument is a strawman of possession.

I'm wondering why you consider this to be wrong.

As far as you're concerned, I did not imply that you're an AnCap but merely a propertarian (i.e. you are willing to accept a stateless society with private property and by extension wage-slavery and rent) and that you did a specific strawman when you were positing what "Communists fail to realize". The whole point is that you do not understand how the tension of property is warranted because you stick to your own definitions rather than understanding what socialists mean by the word.