Question about "Permesso di Soggiorno"

I would like to kindly ask a simple question to You.

I am a student from Mongolia and I have studied in the Italian Language School “Scuola Leonardo da Vinci” in Milan, Italy for the period of 6 months between February 1 to July 23, 2016.

Before I came in Milan, I have applied the Italian Visa in Beijing and obtained the study visa for language schools. Then, in this past July, I fully accomplished my intermediate course and got "B2 level" certificate. So, in order to increase my level and knowledge of Italian language, I wanted to continue my studying and I decided to register another language school which also locates in Milan but more effective and comfortable.

As I mentioned above my previous visa and Permesso di Soggiorno expired at same day July 23, 2016 and I had to extend my Permesso di Soggiorno. Before that expiration I re-submitted my documents and request for renewal of Permesso di Soggiorno to the Police Office (La Questura) on 29 June, 2016. Then, according to my request, the Police Office confirmed the appointment on 12, September, 2016.

Subsequently, I went to the Police Office at this time and I explained my situation that I am continuing my language course and willing to extend my permission.
But the Police Office refused to extend my permission during the appointment.
They clarified that, with my previous letter of Ambassador I cannot extend it without going back to Beijing and for that reason if I would like to extend my permission "I have to return to Beijing and re-apply new Visa".
Am I already overstayed in Italy since 07.27 or within 60 days of expiration I can stay here in Italy?
And *if I have to re-apply a new visa by my own, within how many days I should exit from Italy