There are no such things as "street fighting" martial arts. Each martial art has its own story for how it came to exist and how it has evolved over the years.
Wing Chun kung-fu, for example, is often called a "street fighting" art, but it is nothing of the sort. The founder of that art had a specific purpose in mind for it, and that purpose was to allow someone to quickly learn a style of kung-fu that exploited the weaknesses inherent in traditional kung-fu arts that existed at the time and place where Wing Chun came about. It wasn't for "street fighting".
I explained Wing Chun in more detail in my answer here:
Defence against Wing Chun
Instead of "street fighting", you might have better luck looking for martial arts that are for "self-defense" vs. those that are for the "battlefield".
Battlefield arts are those that rely on sword, spear, shield, armor, and the bow and arrow. They involve strategy that often relies on the fact that you're going into battle with lots of others, in a troop formation. The range is longer distance, and hand-to-hand combat is less emphasized. Some of the weaponry and tactics even deal with people who are riding on horses.
Whereas, self-defense styles have an emphasis on solo hand-to-hand combat either empty handed or using common weaponry (knives and sticks). It's not for the battlefield. It's just for everyday, common situations that require self-defense.
Some kung-fu arts and probably most classical Japanese ryu mix both of these into a single martial art. In some kung-fu arts, they have "village defense" styles, which is kind of like fighting in a militia. But these arts are quietly disappearing in modern times.
"Street fighting" is a phrase the conjures up images of rival gangs fighting each other on city streets. The fighting style these gangs use is not formalized. It's just a collection of skills each gang member learned on his/her own. That's not to say these gang members are bad at fighting. Some are very good at fighting, just because they themselves get into fights a lot. To stay alive, they have had to learn to be brutal and aggressive. Trained martial arts experts often don't have any experience with this kind of environment and the intensely violent and aggressive nature of street fighting, so they often lose in this situation.
Now, "self-defense" based martial arts styles are everywhere. They all attempt to address the same issue: common, everyday situations that require self-defense. These are universal situations. You'll encounter them in every culture and in every time period. And they're not the same as street fighting.
For example: Someone angrily walks right up to you, gets in your face shouting at you, waving his right finger at you, and grabs your shirt at the lapel. This is an intense situation, because he has already crossed the line by grabbing you and getting right up in your face. At any moment he'll probably make a fist with his right hand, wind up, and try to punch you in the face.
Another example: Someone comes up behind you without you seeing him and puts his arms around you, squeezing you as tight as he can while trying to lift you off the ground. He wants to crush your ribs, and he wants to throw you onto your face, head or neck.
Another example: Someone grabs your wrist, twists your arm behind you, and then shoves a knife to your throat. He's behind you. Your arm is twisted behind you also. And he has the knife blade against your throat.
These are all common situations that occur in every culture and every time period. They come about naturally wherever people meet. And self-defense based martial arts all have their own answers to each of these situations.
That being said, there certainly are some "unusual" styles of martial arts out there. Taekwondo and Tangsoodo, for example, were first seen as pretty unusual, because those styles prefer kicking to punching. Ninjutsu is somewhat unusual, because it involves a lot of subtle trickery, mind games, and evasion skills. Aikido is also unusual in many ways, not just because it moves so fluidly and circularly, but also because its main goal is not to seriously hurt ones opponent.
There are all kinds of martial arts out there. I'm glad you had a chance to look into Indian martial arts. It is thought that Chinese kung-fu stems from or was influenced by Indian martial arts.
You should also check into Indonesian martial arts (Silat). And Filipino styles (Kali). There are also Vietnamese styles (Vovinam). Native American Indian martial arts are also pretty interesting, and rare. Taoist based martial arts (such as Bagua, Hsing-Yi, and Taiji) are quite deep in their theory. There are African fighting arts as well. In Turkey, there is Turkish oil wrestling. Etc.
There are also "modern" martial arts styles like MMA, the U.S. Marine Corps combatives training course, and Krav Maga.
This looks like an interesting book on the subject, although I haven't read it:
There have been many books like that which list and discuss many different martial arts. I think they're useful for anyone who's just starting out. It will give you an idea about the philosophical basis of each art and some of their tactics and skills. Some of them might appeal to you. And then you can go out on Youtube to see them in action. From there, you might find someone near you who will teach you.