This invalid use of the term "Israeli" is quite prevalent. There was already a newspaper headline that referred to a "romance between a Druze officer and an Israeli [female] soldier." This Druze officer serves in the Israel Defense Forces, not in the Syrian army. Should he be informed that he is not an Israeli? As is well known, Druze officers define themselves as Israelis more readily than Bedouins in the Negev. In any case, from the perspective of the Jewish majority, terminology suggesting that a group of the state's citizens is not "Israeli" should clearly be rejected. Proper semantics do not solve substantive problems, but poor semantics can definitely be detrimental.
One can assume that this is generally an insensitive habit of speech of a majority that regards itself as the general public, as a majority tends to do. There are surely those who are not prepared, in principle, to recognize that a non-Jew can be an Israeli. However, it seems that more often the problem exists in the opposite direction among those who prefer to call the national group that constitutes the state's majority "Israelis" and not "Jews."
"Israeli" sounds to them like a civic and secular term, while "Jew," in their view, is an ethno-religious definition. I have heard on numerous occasions, from people who are definitely enlightened, sentences such as: "I have Israeli and Arab students in the class." When asked "Aren't your Arab students Israelis?" the answer is always, in clear embarrassment: "Of course, Jews and Arabs." There is no intention here of claiming that Arabs are not Israelis, but rather an intention, or habit, of refraining from defining the Jews as Jews. After all, as noted, an enlightened, liberal and secular person in our times has learned to recoil from the term "Jew," which emits an odor of extreme nationalism and religious coercion.
Thus, the tendency to speak about the "Israeli nation" instead of the "Jewish nation" produces an absurd result, opposite the one intended by proponents of "Israeli nationalism." If the Jews and Arabs in Israel would see themselves as members of the same nation, the correct name would obviously be "Israeli nation." But when both the Jews and Arabs agree that there are two nations, when the Arabs proudly declare that they are an indigenous national minority, it is clear that if the majority is defined as "Israeli nation," the members of the other nation are not Israelis.
It would be best to leave the term "Israeli" as a civic label that applies, in principle, to all citizens of the state. Most people belonging to the majority do not care if they are called Jews, Israeli Jews, or "simply" Israelis: They regard these terms, in most contexts, as synonyms. From the perspective of relations with the Arab minority, it is actually better to call the members of the nation that comprises the majority "Jews" (or "Israeli Jews"), and to call all Israel's citizens, regardless of religion and nationality "Israelis."
But isn't the term "Israeli nation" preferable from a secular perspective? This is an illusion. No advocate of religious coercion would find it difficult to formulate his demands using purely "Israeli" terminology - in the name of the Torah of Israel, the religion of Moses and Israel, the spirit of grandfather Israel, and all for the sake of the people of Israel, of course. As noted, there are no semantic solutions for substantive problems. But it is always possible to make things worse.