in italy the curve for new cases is actually the one in the first graph (except it doesn't rise at the end...yet)

What are you talking about, exactly? The rise in new daily cases, or the wrong perception about them?
I'm from italy, we had a 3 months total lockdown, now since a few weeks rules have relaxed, people are acting as though almost nothing happened, they are *very* relaxed. I was fearing the cases would skyrocket again, but it is not happening. I'm still very cautious about being optimist, in the back of my head I'm still worried but it does seem the situation here is sorta kinda under control
And about how people percieve it, it seems most think the pandemic is just over, even though we still have 300 new cases each day.

Another problem is that I don't know how new cases are identified: since if you test positive, you get put into at least a two week quarantine period where you can't leave your house, as far as I know many people are offered tests but refuse them, which doesn't help the virus to be kept under control :/