You are conflating physical control of something with control of copyright.
The right to delete or modify content isn't the same as copyright. If you write a Wikipedia article, you own the copyright. You license the copyright with a non-exclusive, royalty-free license (Creative Commons) and you put it in a form that practically invites deletion and modification. But that deletion and modification doesn't change your ownership of the original copyright.
On a much simpler level, if I buy a current pop music album on CD, I have the right to "delete/modify" the content. I could put the CD through a wood chipper or drop it from a very high building or any number of other things. None of these would change the copyright status of the album.
> If you post a work on a public site, without explicit copyright, are you granted an implicit copyright?
No. You grant whatever copyright the terms and conditions say you grant. If there is no explicit grant, there is no implicit grant either.
That said, my understanding is that when you write something that's eligible for copyright protection, it is automatic: you don't need to say "Copyright" explicitly at all. It doesn't matter how you use the work in question (although the protections for "unpublished" work are stronger than for "published" work).
Now, it sounds like one element of this trial involved the license implicitly granted to Twitter by posting there. But the very fact that licensing was an issue demonstrates that copyright was in force here. (And the trial evidently concluded that Twitter's license is not tantamount to placing content into the public domain.)