Using the single term Judaism to describe the various religions of the Jewish ethnic / racial group makes as much sense as using "Italianism" to lump together the religions of the Etruscans, the Indo-European pagan religion of the Romans, Mithraism, early sects of Christianity, Catholicism, and so on.
You wouldn't notice this unless you bothered looking into the history of religions founded by Jews, which I've finally gotten around to. I'd always assumed that there was just Judaism, then Christianity building from that, followed by the non-Jewish but still Semitic development of Islam. Old Testament, then New Testament, then the Quran.
I'm ashamed to admit that I had little idea when the Talmud was written down -- circa 500 AD -- even though I dominated quiz bowl in high school and could've told you that Buddha lived before Jesus, that Mahavira founded Jainism, when the Council of Nicea was and what it was about, etc. etc. etc. Everything I'd read, not to mention all that I've heard spoken about it, kept secret the off-shoot nature of what is now called Judaism in everyday language -- that its distinguishing sacred text, as well as the beliefs and behaviors, originated in the Early Middle Ages, not around or shortly after the era of the Old Testament, like I'd assumed.
I repeat that I never encountered this level of obfuscation when it came to any other religion or sect. Every source told you when Mohammed left Mecca for Medina, when Martin Luther nailed the 95 Theses on the Wittenberg Cathedral door, roughly when The Analects and the Tao Te Ching were written, that the Rig Veda came long before the Upanishads, bla bla bla. Yet none of them made it a point to say, "Talmud written down c. 500 AD, ushering in Rabbinic Judaism, the form followed by Jews today." The only date and figure attached to the post-Old Testament history of non-Christian Jewish religion is for Maimonides (and maybe Martin Buber).
Rather than perpetrate the confusion about terminology, I propose we don't use "Judaism" either alone or modified, just as we wouldn't use "Italianism." "Rabbinic Judaism" would require us to call Christianity "Jesusite Judaism" or "New Testament Judaism". For the religion founded with the Talmud, we should just call it Talmudism, Rabbi-ism, or something else that is transparent and that makes no reference to the Jewish ethnic group.
I don't know what to call the collection of sects from the Second Temple period -- Second Temple-ism! Just so long as it doesn't have "Judaism" in it, as that would only make people think that back then there were full-time specialists called rabbis who read a text called the Talmud. The age of the Hebrew prophets that gave birth to the Old Testament, we could call Yahweh-ism. For the time before strict monotheism, use Elohimism.
Why is there a unique deliberate effort to hide basic facts about when the religion followed by today's Jews was founded? This comes from both Jews and non-Jews, so it isn't just the in-group concealing things from the out-group. And it cannot be for fear of somehow derogating the religion by noting that it's more recent than you might have thought -- that would apply to any "newer" religion. Yet no one keeps mum about when Islam began, or when the Lutheran or Mormon sects within Christianity began.
Jews are obviously on board here so they can claim bragging rights -- they were here first, and Christianity and Islam are just spin-offs of their religion, whether they see them neutrally or as corruptions (like most spin-offs are). Even the minority of Jews who are friendly toward Christianity and Islam would still tend to say that you should still be grateful to our religion for providing the foundation of your own. Has anyone ever read or heard a Jew say that their religion, along with Christianity and Islam, are "just" spin-offs of Yahweh-ism, let alone admit that theirs was spun off after Christianity?
You might think that atheist Jews would be the most eager group to get the message out -- talk about the chance to debunk a seriously wrong and seriously popular view about religion, and from insiders no less! But experience tells me that their ethnic pride would over-ride their urge to demystify religion.
Why the non-Jews play along too, I don't know. The obvious first guess is something about the 20th C. effort to reach out more to other religions, inflate their egos, and to do so for Jews in particular as guilt or embarrassment over the Holocaust. But that would mean that before then, say in Victorian or Elizabethan times, non-Jews knew the score and were open about it. I don't know much about the history of non-Jewish perceptions of Talmudism to say, but that's not my impression.
Certainly they looked down on it, maybe thought it was superstitious and overly legalistic, but I don't recall reading European authors saying that Talmudism was a late-comer, after Christianity. Perhaps they just weren't interested enough to look into the history of Talmudism to know when it began?
To end on a suggestion for future research, as they say, someone could do a very simple study to put some meat on these observations, and try to answer some of the why questions. Ask people a question (perhaps mixed in with other irrelevant, distractor questions) about the timing of Yahweh-ism, Christianity, and Talmudism, just a simple ranking task. Forget the wording for now. Let's at least see if people really are so clueless about Talmudism coming last, and if so what fraction of people are off.
Get the standard demographic information, too. I'll bet that the number of years of education is inversely related to getting that question right. Less educated people will probably just guess, while those who've absorbed the message of the elite will be biased toward giving Talmudism an earlier date.
As a control, give them a similar question about Yahweh-ism, Christianity, and Islam. Or throw all four into the same question. I doubt most people, certainly most educated people, would have trouble placing Islam after the others. Again that would mean there's something unique to our (mis)understanding of Talmudism, that it's not simply about it being a religion that we're not so familiar with.
If the rest of the questions are all about religious knowledge, factor analysis will show there to be a single underlying variable, like "general religious knowledge," that makes you more likely to know facts about all religions. However, I suspect that the question on Talmudism would "load" very low on this general factor, and maybe even inversely. That is, knowing lots of factoids about religion would actually hurt your chances of getting the Talmudism question right.
If set in this broader context, someone could actually do this study without raising an eyebrow beforehand, and just comment on it in passing during the write-up. Then let the commenters make a lot of hay out of the results.