Since you have a long commute, we'll recommend bikes in the 125cc segment as they will be quite fuel efficient, which will be an important factor for you. From Honda's stable, you can take a look at the Shine 125, SP125 and Hornet 125 - all three are based on the same platform, but the Hornet is sportier and has more features than the other two. 

From TVS' lineup, you can take a look at the Raider, which will be light, fuel efficient and comfortable and is available across a wide price band too. 

Take proper test rides of all options you shortlist before you make a decision. 

Considering you have some back pain, we would recommend you stick with a motorcycle as the big wheels, longer-travel suspension and neutral riding ergonomics will keep you shielded from road imperfections. Inside your budget, we would recommend 125cc bikes like the Honda Shine 125, Hero Super Splendor / Glamour models, TVS Raider 125 and Hero Xtreme 125R. Depending on the sort of style you are after, you can choose between any of these. 

Since you have asked us about a scooter, our current top picks are the TVS Jupiter 110, Suzuki Access and Honda Activa 125. While the scooters will be more practical, they will not be as ultra comfortable as a motorcycle over bigger bumps. 

Take an exhaustive test ride before you decide.