If you hadn't mentioned "tiny frame" and back pain, I'd agree with mgb about minor changes to your existing bike instead of getting a new one. It sounds like you need a bike that fits you and you have a bike that's too small. There might be a few things that could be adjusted to make a too-small bike work better, but it's likely you're better off getting a new bike that fits right.
We had a similar (not duplicate, I think) question a while back asking about a very similar round-trip distance, etc, you might find reading the question, answers and comments there helpful: What bike+equipment for a long daily urban commute?
A "hybrid" is a bicycle with smooth (road) tires and flat (mountain bike) style handlebars. There's a lot of subtypes that are also "hybrids" such as "commuter", "urban", etc. A lot of them are actually very similar to each other.
From what you describe, I think a good option for you would be a basic "commuter" with smooth tires, flat handlebars and the option to easily add a rack and fenders. It will have handlebars like what you're used to, but roll better (and faster) on roads.
I have a similar length commute and went with a "touring" bike. (basically a heavy duty road bike designed to be able to carry plenty of cargo).
The lower-end "cyclocross" bikes can make really good commuter bikes once you replace the tires. Cyclocross bikes are like road bikes with knobby tires (and often disc brakes). The lower end ones often have the stuff for mounting a rack and/or fenders on them.
Some things to note:
- Consider drop handlebars (road bike) because the choice of handlebar positions can help keep your hands from getting tired on long rides. If you don't like them don't worry about it, though.
- I would avoid any kind of suspension (some "comfort" hybrids have some). They don't do you much good on roads and on a cheaper bike it will just add weight and add one more thing that could break. Stand on the pedals with elbows and knees slightly bent when you see rough patches ahead instead.
- A big heavily cushioned seat can be tempting, but will tend to press into the muscles that you're using to pedal, press into sensitive areas, and rub against things. A lightly padded seat that's just wide enough that you're sitting on it with your pelvic bones will be better for long rides even if it takes a little while to get used to it.
- Try out the bikes. Most shops will let you try a quick ride around the block or something like that.
