The elephants employed by Hannibal were North African elephants, a population that no longer exists. The North African Elephant, Loxodonta africana pharaoensis, is generally considered a subspecies of the African Bush Elephant, Loxodonta africana. Some mammalogists believe that it should be considered a separate species altogether, but this is controversial due to the lack of information we have on the population. North African Elephants are generally believed to be more docile than their southern relatives, explaining how the Carthaginians were able to use them as war creatures. The population is believed to have complete extinct sometime around 100 CE for their ivory.

However Hannibal's personal elephant, an individual known as Surus which translate to "the Syrian", is generally believed to a Syrian Elephant, Elephas maximus asurus, a subspecies of Asian elephant, Elephas maximus. Syrian Elephants were recorded as being the largest of their ilk, a height of 3.5 meters at the shoulder. This intimidating size makes it easy to see way it would be of interest to Hannibal to personally ride. Surus is deduced to be descended from elephants taken by the Diadochi in their campaigns in the Middle East. The Syrian elephant went extinct around the same time as the North African population, once again due to ivory.

So when you ask which type of elephant Hannibal availed, the most likely answer is, surprisingly, both, but at the same time neither as the elephants were of different subspecies than extant ones.

A quick word on Indian vs. Asian elephants. The Indian Elephant, Elephas maximus indicus, is merely a subspecies of Asian elephant, the latter of which compasses island subspecies in Sri Lanka and Sumatra, as well as the extinct Syrian population.