So I hope I did not misunderstand what most sources meant when they say Inari Okami worship is personalized, but:
My take is that somewhen in the distance past (seemingly 5th-7th century AD?), there was an original deity Inari. As history and thus the need of their worshipper changed, the deity itself changed to incorporated those need.
Thus, again as my personal understand, the development of Inari worship have a marked difference from, for example, some Egyptian deity who start out as two distinct deities, but as their "function" become overlapped, got merged into one.
So my question is, if I understand correctly, how did we know that Inari "branch out" from an original deity, and not multiple distinct deities that "merged" into a single one called Inari, especially since to my knowledge, many facets of Inari ARE called by distinct names, and in the Fushimi shrine complex there ARE multiple kami being housed there?
The short answer is that you can't be sure about anything before the Edo period. The more detailed explanation follows...
First off, it's important to note that when people talk about Shinto or Buddhism, they're talking about Japan. These religions were imported from China during the Tang dynasty at the end of the 6th century CE. They became popular because of their similarities with Taoism, which had been around for centuries, so these imports are often referred to collectively as Chinese religion.
When this happened, a lot of existing culture was absorbed into the new system. For instance, the old gods of war and agriculture were reinterpreted as channelling aspects of the emperor himself, rather than having independent power. There's also evidence of similar changes happening elsewhere, such as Korea.
It's well known that the Emperor Jimmu unified Yamato sometime in the early 7th century, and he founded the first capital city at Nara. This means that any records from before then would be highly suspect. Without proper documentation, it's impossible to tell whether something existed before or after Jimmu.
That said, there are still plenty of ways to date things. One way is through archaeological excavation. Another is through written documents.
Archaeological excavations are only possible on certain sites, however, so they aren't very useful here. That leaves us with the written record.
There's no shortage of historical texts, but few of them actually mention either Inari or Enmei. Of course, it could just mean that they weren't prominent enough to warrant mentioning.
One exception is the Kojiki, the oldest extant work of Japanese literature. It was compiled between 806 and 825 CE, which puts it firmly within the range where we might expect to find mentions of Inari.
Unfortunately, while it does mention Inari, it doesn't give much information about her (or him). She simply appears along with other gods, and she has no personality traits described.
To get a sense of what kind of god Inari was supposed to be, we have to look at other works. Unfortunately, most of these works are too modern to serve as good references for dating purposes. However, we do have one major piece of writing from the time that describes her in some detail, and it gives us a pretty good idea of how people thought about Inari back then.
This is the Man'yoshu, an anthology of poems composed in classical Japanese style. The compiler of this book was Fujiwara no Teika, who lived in the late Heian period, and the collection contains over 1000 pieces.
The last section of the Man'yoshu is called the Hyakunin Isshu, and it consists of 100 short stories. Each story is named, and each one begins with the word 'Nanuki', which roughly translates to 'little snow'.
The Nanuki stories contain a lot of information about Inari, but almost none of it is about Inari herself. Instead, the focus is on the little snow itself. Inari is mentioned in the context of snow falling, but that's all.
The main thing we learn is that Inari is associated with the mountains. The stories talk about Inari being born in the mountains, and the names of Inari shrines often include the words yama, tsuyu, and kami. This suggests that Inari is associated with the mountain and the sky, and that the worship of Inari is focused on the snowy landscape.
In addition to being linked with the mountains, Inari also seems to be connected to the moon. The Nanuki stories describe Inari as being born under the full moon, and there's even one tale in which the moon is used to predict the future.
Another important aspect of Inari's mythology is that the gods are sometimes referred to using feminine pronouns. The Nanuki stories use the pronoun nani, which can be translated to mean either'she' or 'her'. This may suggest that Inari's gender is ambiguous.
Finally, there's a story in the Nanuki about Inari eating the heart of the sun. While this is a common theme for Japanese folk religion, it's not really a part of Shinto. It isn't clear whether this is a reference to a real event, or if it was just a poetic flourish.
With this information, it's possible to draw a few conclusions. First, Inari was originally a goddess of the mountains and the skies. She was probably associated with the moon, and she had a close connection with the sun.