exile

What is complete redemption? If the Rabbi means that there are still troubles for the people of Israel, then there were troubles even during the time of the Temple and yet the exile was only considered when we were exiled from our land? Or is the reference to the Temple?

Temple, repentance, enslavement of kingdoms

In other words, even if we return to what we were before the exile, namely the Temple with the enslavement of kingdoms and no repentance, it still would not be considered complete redemption?

During the days of Jeroboam, most of Israel served the Jews, and when the Jews were under Cyrus, they were also in exile.

You have to live through a situation in order to take a stand on it. Such a return is not like the situation before the exile. That they were sinners is always true, but today we are not talking about sinners but about people who have neither Judaism nor faith nor anything. It is a completely different situation.

This sounds like something undefined. The question is whether, from a halachic perspective (for example, regarding the change in the wording of the prayer asked above), there are clear definitions of what constitutes complete redemption?

Indeed, it is not completely defined. But you are making the question difficult and the onus of defining it is on you, not me.

The kings of the First Temple period had less to do with Judaism than we do today. They had a different god with a different temple, a completely different religion, their own holidays, etc. The modern state of Israel observes the Torah in its most extreme (Orthodox) interpretation. All government institutions observe Shabbat. They have laws for Yom Kippur, chametz, and kosher. The government funds the rabbinate, synagogues, yeshivots, and mikvahs. What do you want them to do, forcing everyone to keep the commandments. Something that has never existed in the history of the Jewish people?
Besides, how do these things relate to redemption? See Shabbat 33:1, that the Temple and exile are two separate things. Obedience to the commandments is the reason for exile, not its definition. And why do you think we still have royal servitude? We are certainly less enslaved to the Gentiles than we were in the days of the Temple. In my opinion, the definition of redemption is very simple, “to be a free people in our own land”