I cannot be certain but if the app says 'cerificazione valida in Italia e Europa' with the green tick it does translate as valid.
I was in Italy last November and never once was asked to show a pass. I am travelling to Italy again next Saturday for a few days and will just have my British NHS app.
Sorry I can't be of any more help.
Have a great time in Italy, eating in Venice can be expensive for touristic below par food, I recommend finding these places here. https://www.veneziaristoranti.it/it/ristoranti.html61 They are all good, especially the antiche carampane. https://www.antichecarampane.com/it/171 These are not places to get pizza or lasagne, wait till you get to Rome for these! After sightseeing in Rome Trastevere is a lively place to stroll around with plenty of decent restaurants.
all the best.