There are a lot of potential definitions of "public debt," and differing statistical agencies use different ones. (They may have more specific names, such as "Federal debt held by the public.")
A lot of these definitions exclude debt held by the central bank, based on theoretical arguments that may or may not make sense. (These debt holdings would roughly correspond to the money supply, which is allegedly not "debt" according to some economists.)
Is any particular definition "correct"? It seems that the "correct" definition would be practically useful. However, as Professor Reinhart discovered the hard way (the "90% debt limit paper"), government debt levels (however defined) do not appear to offer much predictive power for economic outcomes. If this is indeed the case, why worry about what definition is used?
Arguing that ECB "debt" is meaningful implies a belief that the ECB can default on euro-denominated debts. It is very unclear how such an event can occur, other than in the context of a disorderly break up of the euro area. However, that would imply the end of the existence of the euro itself -- in which case, why care about euro-denominated debt values? It will all have to be defaulted on anyway.