No, you cannot drive electric scooters without license legally. Before 2016 it was allowed to use electric(!) scooters with less than 250W power and 25km/h top speed, but those days are over.
The idea that scooters are exempt is quite understandable because it really seems easy to lend scooters even without license or helmet and because traffic laws in India are seemingly ignored by the majority of the population.
As already said in the comments, I absolutely advise against driving any vehicle (scooter, motorcycle, car) in India in the urban cities and the main roads. I searched the stats for deadly vehicle accidents: In 2013 Germany had 52 million vehicles and 3 540 deaths caused by vehicle accidents. Also at 2013 India had 183 million vehicles (Factor 3.5 more than Germany), but 238 000 deaths! (Factor 67 more than Germany), making India approximately 20 times more dangerous than Germany to drive. I admit I simply don't understand the mindset of people who think they could drive the most dangerous parts of the world without any experience at all. It is like a person who barely can stand on skiers who tries to make ski jumps.
Given that, if you are in remote areas, far away from the main roads, I think it is tolerable to discover India with a scooter/motorcycle/car and a valid driving license. Still the roads must be driven carefully (many potholes, cows, chicken and slippery dung on the road), the weather (monsoon!) can be quite impressive or even dangerous and the people seem to be completely oblivious to vehicles, walking or crossing the road like they want.