Second, a ‘track to constraint’ makes an object axis poit to another object center, this is the correct behaviour
Now
If you really want to move the Object center you shoud place the cursor (The red wite circle) where you want your center to be, then in Editing Context Mesh Buttons in ObjectMode in Mesh panel you can press ‘Center Cursor’
THe other button ‘Center New’ places the center in the middle of the object.
This doesn’t clear the constraint. It did at first, but I removed it (it was never released that way). To remove a constraint, go to the Object window (F7) and click the X button on the constraint block you want to remove.