More official links:
For the 060608 site (in English)
Taxi - Info, fares, infoline
From that site it says:
HOW DOES IT WORK
060609 Call a taxi operates through a voice recognition system that recognizes street names. Once you have dialled the number (+39 060609), just follow the instructions: clearly indicate the pick up address and the system will redirect the call to the nearest taxi rank, where a taxi driver will answer your call.
But here's 060608's page showing the individual taxi companies with their individual phone numbers/websites
Transports » Getting around » By taxi
Note regarding paying with credit cards:
On that page directly linked above I clicked on a handful of those individual taxi company names highlighted in orange and saw a large majority saying "All Credit Cards Accepted" - so perhaps it would make sense to call one of those companies directly when using credit cards -