Object has a name and contains the data. The data has it's own name and can be linked to multiple objects.(Linked duplicate) In your example the data type is armature. Then each bone has its own name.
" the name tickbox in the data tab names the bones, whereas the tickbox in the object tab names the armature?"
That is correct. The Name in the Object Properties Tab shows the Object's Name and that is in the case of an Armature, the Armature.
(You can also hover over the tickbox, or the word Name and look at the Tooltips (they should be enabled by default); they can be very helpful sometimes.)
Cheers guys. As much fun (and pain lol) I had with creating the models, I feel like I'm finally up to the exciting part with rigging and animation!