Just to clarify, there is no such thing as an IDL (International Driver License). There is an IDP (International Driving Permit). The IDP is a translation of your driving license from your home country - it is not a license in itself. When you use an IDP you must also carry your actual driving license as well. You are only endorsed to drive whatever you are licensed to drive on your driver's license.
So, if you didn't get an IDP in your home country, you can't get one anywhere else, as no other country is allowed to issue a translation of your driver license.
There "tourist driving license" issued in Bali is for Bali only. I also am not aware of an equivalent arrangement outside of Bali.
You might be ok using your Greek driving license, in that it might be accepted by the police and hire car companies if they check, however do check your travel insurance, as many insurance companies get picky about your license status when you are in control of the vehicle.
For the most part, in Lombok, it is not expensive to hire a driver with vehicle - so that is the best way to go if you want to tour the island.
If you want to hire motorbikes, apart from the insurance issues, it is pretty easy to get away without having an IDP - and just cop a small fine if you are caught.