That is my impression. He may have also killed Dumbledore, but a murder was committed, and that was the requirement for making the soul splitting thingy.
Can anyone say Star Trek III?
The singing Phoenix had me thinking that he was shedding healing tears...Dumbledore suffered an injury from destroying the ring, he could have suffered an injury while destroying the locket. I don't believe for a minute that DD was fooled by some fake locket.
Also if Dumbledore has gone into hiding then it makes sense that his Phoenix would also leave and go with him. DD being presumed dead makes it easy for Snape and Malfoy to be accepted into Voldermore circle. DD very own sleeper cell if you will.