What defines a "Jewish State" according to Rav Kook?

Question

This question is only according to the opinion of R' Kook, who says that modern day Israel "would be considered the quasi-Kingdom of Israel".

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What component of the modern-day state would give it that distinction?

1. Is it the amount of Jews living there? (If so, is it the percentage of country being Jewish vs. percentage of the Jews living in the country)?
2. Is it the general territorial consistency with Biblical Israel? (As in, most of modern-day Israel is inside Biblical Israel).
3. Is it that the majority of the rulers/government officials are Jews?

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EDIT

(Somewhat) practical ramifications:

1. If Israel would have been made in Uganda, would R' Kook be of the same opinion?
2. If Jews would flock to (for example) Lithuania, and the majority of Lithuania would be Jewish (and the government, being democratic, would probably have a Jewish majority), would R' Kook consider Lithuania "a quasi-Kingdom of Israel"?
3. If, for example, the story of the Khazars is true, the King and nobility converted to Judaism. Would he have the same law as a classic Biblical King?

Answer 1

According to Rav Kook, and his son, Israel today is considered the quasi kingdom of Israel for many reasons.

I list them here in no particular order.

1. With the signing of the Balfour Declaration, the "three oaths" have been fulfilled, and the Nations of the world (like in the time of Cyrus) told the Jewish people that they are to return home and build their state.
2. It is in the land of Israel, who's borders over various Jewish kingdoms grew and shrank.
3. In Israel today, it is the first time that Jews report to no government other than a Jewish government. There is no other nation ruling over us.

4. The gemora says that the clearest sign of the time of the redemption is that the hills of Israel will bloom with fruit. The first stage of that redemption is the in gathering of the exiles to Israel. This happened at the time that the Balfour Deceleration was signed.

5. The Rambam (according to Rav Kook, I havn't seen the Rambam myself here) writes that the future kingdom of Israel will happen in a few stages. First, the Exiles will be returned. Then, when they are returned they will conquer all the land of Israel. Then or during that time, the Anshei Keneset Hagadol (120 people) will be re-established so that they may choose a "king". Then, when all the land that it had in the times of Joshuah is conqured, then, the "king" will e rebuild the beit hamikdash. So while we have not yet conquered all the land, we only have a Qausi-Kingdom.

According to this view, the Milchemet Mitzvot will not be done under a purely halachic leadership/state, as that will only come when the King is established. And when the king is established will only be after the wars have been finished. This is why the moshiach will be able to turn the swords into ploughshares. Ofcourse the State of Israel should follow halacha as much as possible, but it is not a requirement until the SanHedrin and King are established, and the temple is rebuilt.

It is noted by Rabbi David Bar Hayim, that the mitzvah in the Torah of "Melechet Kohanim", would be translated today as "Midinat Kohanim", as during the time of the Torah's writing, a "Medina" only meant a City-State and not a nation. Thus establishing a state run by Halacha is itself its own mitzvah. However, Rav Kook felt that the importance was only in the Klal and in all the Jewish people working together and being in self rule. Whether it be a secular, democratic, religious, fascist, or any other type of government as defined by the Jewish people, and agreed upon by all the Jewish people.

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In response to your edit:

1. Rav Kook was against the Jews living in Uganda. He would not have seen it as a Jewish State, but rather just another place where Jews live in Galut.
2. No, it's not the land of Israel.
3. Rav Kook said that learning the book of the Kuzari was the most important sefer to learn in regards to Emunah. However, he never made any mention of them being an example of a Jewish government or a Jewish kingdom. Since he never mentioned it, I would have to assume that no, he did not view the Khazar king as having the status of a Biblical King. But perhaps based on his logic one might say he was. Seems like it's own interesting question.

Answer 2

Rabbi Hershel Schachter points to the Gemara's observation that King Cyrus of Persia, the son of Queen Esther and King Achashverosh, has his years on the throne counted in some accounts from 1 Nissan (like Jewish kings), and in others from 1 Tishrei (like non-Jewish kings). Though he was the king of Persia, as long as he was sympathetic to Jewish causes (he basically gave Ezra and Nechemiah a blank check to build the Second Temple), that made him a "Jewish king." Later in life he "went sour" with his Jewish identity (achar she'hichmitz), at which point he was counted like a non-Jewish king.

Rabbi Schachter uses that as proof that some sort of Jewish identity or sympathy is sufficient for today's State of Israel. You can draw your own conclusion from there to the various hypotheticals you describe.

Of course Rabbi Schachter is discussing the mitzva of som tasim alecha melech, which he defines as: establish a Jewish government over the Biblical land of Israel. His definition of "Jewish government" is the above.