Alpha and Beta Israel

Shalom, As an outsider to the Torah community, I was curious of a certain historical topic which came to mind. Upon researching the Jewish religion through books, I discovered a rather interesting topic regarding this term of "Two Israels" which told about Alpha Israel, talking of Judea, and Beta Israel, which talked of Ethiopia. Rather than researching on the internet for a conclusion, I would rather get an answer from a dedicated source. So here is my question: What are the religious, political, and historical differences between Alpha and Beta Israel and why exactly are they called "Alpha" and "Beta"?

The term "Beta Israel" has only been used very recently since the rediscovery of some 60,000 Jews in Ethiopia, and only since then have some (very few) also begun using the term Alpha Israel in contrast, regarding the rest (majority) of the Jews (about 12,000,000). Most use the general term "Jew" regarding them all, because most of the other 10 tribes of Israel were lost in the exile of Assyria, about 722 BCE, and the vast majority of those who survived to this day, whether in Europe, Asia, Africa, Ashkenaz, Sfarad, Yemen, etc. were those exiled in the year 70 CE from "Judah" (in addition to the Cohens and Levis and some from the tribe of Benjamin which was basically annexed by Judah). The Ethiopian community claims to be from the tribe of Dan (one of the 10 lost tribes), and were basically almost unheard of by the rest of Israel for about 2,600 years. Accordingly, many of their customs are different, but many of the laws and customs have impressively survived all those centuries, albeit not as well as most other communities who have “only” been separated for 1,900 years (and continued some form of contact with each other). The Chief Rabbinate asks these Jews (or more accurately: “Israelis” or “Israelites”) to undergo a formal conversion before they marry, just to "play it safe" and remove even the slightest doubt regarding their familial identity. Almost all of them came back to Israel in the early 90’s in a secret arrangement with the Ethiopian authorities, and despite coming from a “third-world country”, have on the whole been absorbed relatively well, and integrated into mainstream Israeli society. Nevertheless, there has been controversy in recent years regarding the “Falashmura”, those Jewish families converted by missionaries to Christianity over the last century to avoid anti-Semitism, and now wish to come to Israel and return to their original roots, who are resented as “traitors” by the regular Ethiopian Jewish community but are legally accepted by the State of Israel after proving their Jewish lineage.