I am not expert in the Talmud like the Rabbi, but from the little I have learned, usually when the Talmud uses the question “what is,” it is to clarify something that actually requires clarification.
It is troubling that hundreds of years after the events, the Sages needed to clarify what Hanukkah is.
If not for the wording “what is,” fine; I would have expected the Talmud to begin with “Our Rabbis taught,” a statement of fact/story, not the question “what is this.”
How is this different from the fixed text of the Passover Seder that includes “Why is this night different?” which the children ask in order to keep them focused?

The Talmud is full of didactic questions, and this clearly seems to be one of them too. Nobody here was discovering anew what Hanukkah is. Though the miracle of the cruse of oil appears in the Talmud after the destruction, and it is doubtful whether it was there originally; perhaps the purpose was to insert it.
Exactly like “Why is this night different?” A didactic way of presenting an answer.

By the way, in the Talmud too, the answer to “What is Hanukkah?” is a baraita. So it really does begin with “Our Rabbis taught.”

“What is Hanukkah?”—meaning, why do we celebrate it even though Megillat Ta’anit was annulled.

I will have to accept the Rabbi’s words, at least until I fill my belly with the Talmud and halakhic decisors.

As for the miracle of the cruse of oil, this really is apparently the first source.

Since the Rabbi mentioned it, I’ll allow myself to point out two things:

A. If Hanukkah was instituted because of this miracle, that is strange. We do not find any other festival instituted because of a miracle. Otherwise we would have to celebrate the One who told the oil to burn telling the vinegar to burn as well, and recite Hallel in memory of the Shunammite woman’s oil and the arak of the Baba Sali that were endless. This may be the highlight of the event, but surely Hanukkah was instituted because of the military victory and the return to the Temple.

B. Maimonides quotes more or less the baraita of “What is Hanukkah?” with one small/large change: he deletes the words “and a miracle occurred with it.” I do not know how Maimonides would explain that they lit with a cruse of oil sufficient for one day for eight days straight—the gates of answers have not been locked—but he did not like the miracle aspect.

A. We certainly do find that: on Purim the enemy sought to destroy and kill, and a miracle happened and it was turned upside down. On Hanukkah the enemy sought to make them forget Your Torah, and a miracle happened in that they did not succeed even in making the lighting of the menorah be forgotten, by means of the impurity of the oil.
B. Maimonides himself established a day in memory of the private miracle that happened to him on a ship, as explained in his famous letter. It seems more that you do not like a miracle that strikes you as something mystical—but the splitting of the Sea and the ten plagues are not far from that.
Happy Hanukkah.

A. Purim is not about a physical miracle at the level of nature. The victory in Hanukkah is also miraculous—many delivered into the hands of the few, etc.—but in both cases it is not something that contradicts the laws of nature. By contrast, oil sufficient for one day burning for eight days is a change from the ordinary laws of nature.
B. Maimonides’ story about the ship does not include the word “miracle,” but rather personal thanks for coming out of a dangerous situation.
As for the splitting of the Sea and other things, Maimonides’ approach to miracles is well known. See, for example, his commentary on the Mishnah in Avot regarding the ten things created on Friday at twilight, and elsewhere, but this is not the place to go on at length.

The words of Maimonides in his commentary to the Mishnah, Avot chapter 5, where he explicitly speaks about miracles outside nature:
But the ten miracles that were done for our ancestors in Egypt were their being saved from the ten plagues, with each and every plague affecting Egypt specifically and not Israel, and these are certainly miracles… But the ten miracles that occurred at the Sea are a tradition: the first was the splitting of the water, according to the plain meaning of the verse, “and the waters were split” (Exodus 14:21).
The second was that after they split, they became like a vault, until they took on the form of a roof and were not covered over and not slanted, and the path was as though there were a tunnel in the water, with water on the right and left and above, as stated by Habakkuk, “You pierced with his own rods the heads of his warriors” (Habakkuk 3:14).
The third was that its ground hardened and froze for them, as it says, “they went on dry land” (Exodus 14:29), and there remained on its bottom no mud or mire as in other rivers.
The fourth was that the paths of the Egyptians were sticky clay, as it says, “the mire of many waters” (Habakkuk 3:15).
And the fifth was that it split into many paths corresponding to the number of the tribes, in a kind of curved arc in this form, as it says, “to Him who divided the Sea of Reeds into parts” (Psalms 136:13).

Also in the laws of Hanukkah Maimonides speaks about a miracle:
And because of this, the Sages of that generation instituted that these eight days, beginning on the twenty-fifth of Kislev, should be days of joy and Hallel, and candles are lit in them in the evening at the entrances of the houses on each and every night of the eight nights, in order to show and reveal the miracle.
Which miracle is being referred to that has a connection to eight days?

Well said. But you did not look at the words of Maimonides that I mentioned regarding the ten things created on Friday at twilight. The miracles too were created within nature; see there.

Accordingly, Maimonides minimizes as much as possible miracles that cannot be interpreted according to the way of nature. Victory in war by the few against the many is a miracle, but not supernatural. Oil sufficient for one day burning for eight days is against the laws of nature, and therefore Maimonides does not focus on this miracle, whether it happened or perhaps did not.

I did not understand your words, “he does not focus on this miracle, whether it happened or not.” If Maimonides writes that the whole point of Hanukkah is lighting candles for eight nights “to reveal the miracle”—that is exactly what the days of Hanukkah are for, publicizing the miracle, and especially the miracle of the oil, because the miracle of the war is not an eight-day matter, nor a matter of candles.
If Maimonides had any doubt whether the miracle happened, he would not have written the laws of Hanukkah—perhaps only the laws of Hallel.

As for the ten things, thank you very much; I will look into it, God willing. In general, Maimonides’ way is to bring things as close as possible to rationality, and that is also the way of Radak—all this insofar as possible.

Maimonides means the miracle of the victory and the heroism and the return to the Temple, not the miracle of the cruse of oil, where he omitted the Talmud’s words about a miracle.
Of course the cruse of oil is the highlight—literally—of the return to the Temple, and perhaps it shaped the days of Hanukkah and the lighting in homes for eight days, though in other places such as the Book of Maccabees entirely different reasons are given.
As support for this, in “For the Miracles” the miracle of the cruse of oil is not mentioned.

Regarding the question at the start of the discussion,
in my opinion the meaning of the word Hanukkah is clear, as written in the Book of Maccabees: the dedication of the Temple and the altar that had been destroyed and renewed. If so, the festival of Hanukkah is a festival of rejoicing over the Temple.
Therefore, apparently there is no place to celebrate Hanukkah after the destruction of the Temple. So the Sages ask after the destruction of the Temple, “What is Hanukkah?”—how is Hanukkah relevant to our times after the Temple was destroyed?
And the Sages explain that there is no answer on the level of the plain meaning, but there is one on the homiletic level, and therefore they interpreted the word Hanukkah as two words: “they rested on the twenty-fifth.”