All of these are unimportant.
Names (Christianity, Judaism or even Islam) don't make religions.
The core of Judaism, which is the Written Torah and Oral Torah comes from what are taught, not by name. It doesn't matter Moses didn't name his religion, because it is not his place to give name to his religion.
The core of Christianity comes from the what Jesus taught and did, and dozens or so letters from disciples of the 1st century CE, not from name.
The names are meaningless, it what's being taught that count. You can swap with any name you like, and it would be irrelevant, except for you.
Hebrews, Israelites, Jews or whatever you want to call them identified their religion, by the god they worshipped, whether he be called by YHWH, Elohim, etc, not by the name Judaism.
I have studied a lot of ancient religions and myths, and rarely do they give names to their respective religions. The religion was not identified by their names, but by the deities (gods or goddesses) they have worshipped.
I will give you some examples. In Heliopolis, Egypt, they worshipped Atum or Ra, and the Ennead (Atum's children and grandchildren), as far back of the 3rd dynasty of the Old Kingdom. There were no name in the writings. It is modern scholars (historians, archaeologists, anthropologists) that applied the Heliopolitan religion. Even when Aten created a new monotheistic religion for his god, Aten, Akhenaten (Amenhotep IV), there was no "Atenism"; it is modern name for the religion of Aten.
The same can be applied to the Sumerian religion in Mesopotamia. Particular cults existed in each Sumerian city. In Uruk (known as Erech in the bible), two deities of the greater importance were worshipped in this city: An (Babylonian Anu) and Inana (Ishtar). They were important because temples were built to them. No names were given to the cults.
Only Muslims would resort to names being of importance. Just because Muhammad gave name to his religion, doesn't mean everyone have to. It is rather petty.
OK, you think name of religion is important. Then answer some of my questions.
According to Islam, and to Muslims, Islam IS SUPPOSEDLY an eternal religion.
Yes, Moses didn't call religion Judaism or Jesus didn't call religion Christianity.
- Then why didn't God just tell Moses or Jesus to call their religions - ISLAM?
And according to Muslims, all the prophets and messengers were supposedly "Muslims".
- Then why didn't Moses, Jesus or other prophets call themselves "Muslims"?
Your god, Allah, is supposed to be all-powerful and all-knowing, so time would have no meaning to one such like him. And yet why didn't the names "Islam" and "Muslim" exist prior to Muhammad supposedly became a prophet?
They didn't exist because Allah wasn't all-knowing.