A co-worker from Italy told me. But you can find this for example here as well: https://www.aljazeera.com/news/2020/03/italy-south-korea-dif... "Italy started out testing widely, then narrowed the focus so that now, authorities do not have to process hundreds of thousands of tests." (I don't normally read Aljazeera... just found this using Google.) It matches what my co-worker told me. See also https://www.worldometers.info/coronavirus/covid-19-testing/ (I have no clue how reliable that source is)
I live in Italy and I'm not aware of any changes in the procedure for testing.
If I read the worldometer link you provided what I see is that both on March 2nd and on March 9th Italy was second only to Korea in number of tests performed and that we moved from 386 tests per million people to 1005 in the same period. What am I reading wrong?