Not sure how the thread turned into sidereal tropic debate, but that's what I read from JA's reply to jazz.
For Arabic Astrology, should Babylonia not accounted as one too? Babylonia was the ancient land for now Iraq? Surely Iraq is that part of the world?
If so, Babylonians had sophisticated astrology from BC 700-800 time having their own ephemerides, charts with angles of stars, dignity tables etc? Their system had been passed onto ancient Egypt, and then to Greece and Rome?
The "hellenistic world" was the different empires and kingdoms that developed around the mediterranean and part of the middle east, after the conquests of Alexander the Great, around the year 320 BC. They are called this way, because hellenic culture (ancient greek) was spread through these kindgoms and states and mixed with ancient asian cultures (such as that of the persians). Eventually most of these kingdoms fell under the control of the Roman Empire. The astrology from this period of time is what most astrologers refer to as "hellenistic astrology"
Centuries later, during the 7th century AD, the muslim empire conquered a huge amount of those territories, mostly in northern African, Asia minor, and part of Europe. They absorbed part of this greek culture into their own, and astrology was part of that knowledge. The astrology that was developed and expanded within this period is what we usually call as "persian or arabic astrology". The reason why that name is chosen, is probably because not all of the astrologers were actually muslim (for example Ibn-Ezra was jewish living in muslim Spain), so astrologers today settled for the term "arabic" or "persian" (from historical perspective the muslim empire's origin was in arabia/persia)
The original parts were created during the hellenistic period. However, they are popularly called "arabic" parts, because they were expanded under the influence of the "arabic astrologers" of the muslim empire during the middle-ages. The muslim astrologers also created many of them for different uses.
This is a time table of how we divide the periods of time and "schools" of astrology within what we call traditional. The dates may not be super accurate, but more or less:
Before 3rd century BC: Proto-astrology (Babylonia/Egypt)
From 3rd century BC to 7th century AD: Hellenistic Astrology (Hellenic Kingdoms/Roman empire)- use of Horoscopos**
From 7th century AD to 15th century AD: Persian Astrology (Muslim empire) - also called Arabic Astrology
From 15th century AD until 19th century: Reinassance Astrology (Europe)
From 19th century until now: Modern Astrology, in all its forms (worldwide)
** The historical hellenic period officially ends with the fall of the roman republic and the rise of the roman empire, but in astrology both periods are usually considered as part of the "hellenistic astrology" period.
You can also add Indian Astrology as a separate branch that developed on its own in Asia.