What is full redemption? If the Rabbi means that the Jewish people still have troubles, then there were also troubles in the time of the Temple, and exile was still considered to begin only once we were exiled from our land. Or does he mean the Temple?

The Temple, repentance, subjugation to foreign kingdoms.

Meaning, even if we return to what we were before the exile—that is, the Temple together with subjugation to foreign kingdoms and without repentance—it still would not be considered complete redemption?

In the days of Jeroboam, when most of the Jewish people worshiped idolatry, and when the Land of Israel was under Cyrus, were we also in exile?

You need to experience some actual situation in order to determine a position about it. Such a return is not like the situation before the exile. The fact that there were sinners was always true, but today we are not talking about sinners; we are talking about people who have nothing to do with Judaism or with faith or with anything. That is a completely different situation.

That sounds like something undefined. The question is whether, from a halakhic standpoint—for example regarding changing the wording of the prayer asked about above—there are clear definitions of what complete redemption is.

Indeed, it is not completely defined. But you are raising an objection, and the burden of defining it is on you, not on me.

The kings in the First Temple period had less connection to Judaism than we do today. They had a different god with a different temple, a completely different religion, their own festivals, and so on. The modern State of Israel observes the Torah according to its most stringent interpretation, that is, Orthodox. All government institutions observe the Sabbath. There are laws about Yom Kippur, leavened food, and kashrut. The government funds the rabbinate, synagogues, yeshivas, and ritual baths. What do you want—that they should force everyone to keep the commandments? Something that never existed in the history of the Jewish people?
Besides, how are these things connected to redemption? See Babylonian Talmud, Shabbat 33a, where the Temple and exile are two separate things. Commandment observance is the reason for exile, not its definition. And why do you think we are still under subjugation to foreign rule? We are certainly less subjugated to non-Jews than in the days of the Temple. In my opinion, the definition of redemption is very simple: “to be a free people in our land.”