Of course it is more complex than that. I would suggest everyone’s identity in Ireland (and elsewhere) is a lot more complex than that. However, we agree that he was Irish (the country of his and his families birth). You strive to identify his nuanced form of Irish identity. I do not think this is helpful, as this is contentious and can not be simply asserted in the opening paragraph. 

 So, if we have to discuss his form of Irish identity (I think dealt with quite sufficiently in the subsequent section, his protestant background etc) we need to understand that 
 1. Anglo-Irishness generally refers to the wave of “new-english” colonisation which occurred after the 16th century. Clearly Wellington’s ancestry pre-dates this time frame (Online: The Anglo-Irish, Fidelma Maguire, University College Cork). Indeed, the Norman invasion brought his family to Ireland along with other families such as Burke, Fitgerald, Darcy, Joyce, Barry, Clare etc., families which cannot but be declared simply Irish. 
 2. Since the 19th century, the term Anglo-Irish has been used in a prejudicial manner to marginalize a certain section of the Irish population (Online: The Anglo-Irish, Fidelma Maguire, University College Cork). The term is consequently contentious.  
 3. Finally, even for those historical figures whose ancestry does fall into the correct time frame according to Maguire, for example Charles Stuart Parnell, Isaac Butt, John Millington Synge, Oscar Wilde, George Bernard Shaw, WB Yeats, Samuel Beckett (or was he of Huguenot!), they are clearly identified as Irish in the literature and, rather importantly in this case, Wikipedia articles. 
 I therefore see no reason why the same standards and logic should apply and why Wellington should not be included in such outstanding company. Can you?