If you remember the parshiyos from earlier this year (or cheat and use a concordance) I think the answer will be clear. The term “Ivri” comes up again and again in the beginning of Shmos. A few examples: the “miyaldos ha’Ivriyos: (1:15) save Jewish babies, including Moshe, who bas Pharoah refers to as being “m’yaldei ha’Ivrim.” (2:6) Later, Moshe goes out and sees an Egyptian hitting an “ish Ivri” as well as two “Ivrim” who are fighting. Hashem tells Moshe to tell Pharaoh that the G-d of the “Ivrim” has appeared to him (3:18). At this point in history there is no Jewish nation. There is a large family, a tribe of related members. It is only later, post-exodus, after kabbalas haTorah, that we become a nation. Once that happens, the term “Ivri” vanishes. The only occurrence of the term “Ivri” after the exodus is in reference to the Jewish slave. We are now Bnei Yisrael, Am Yisrael, not Ivrim.
Perhaps the Torah deliberately uses the term “Ivri” with respect to the slave to indicate that the slave has forfeited his identity as a “citizen” in the nation of Am Yisrael. He still retains his relationship to us as a people, he still retains his identity as a member of the family/tribe of bnei Avraham, Yitzchak, and Yaakov, the organizational unity of “Ivrim” that pre-dates our nationhood, but he has forfeited his rights and privileges beyond that.
In Parshas Shmos, when Moshe and Aharon first appear before Pharoah, they tell him (5:1) that “Hashem Elokei Yisrael” has demanded the release of his people to celebrate a “chag.” Pharoah responds that he does not recognize the deity they are talking about and therefore won’t agree to terms. Moshe and Aharon then repeat the same request (5:3) using slightly different language, telling Pharoah that “Elokei ha’Ivrim” demands the release of his people to offer sacrifices to him. This time Pharoah throws them out. Why did Moshe and Aharon think repeating the request a second time would make a difference? And why was Pharoah’s response so much harsher this second time?
Netziv explains that when Pharoah heard the term “Elokei Yisrael” he assumed Moshe and Aharon were speaking about letting the spiritual elite of the people go out for a celebration, a chag. Yisrael is the name Yaakov is given only after he manages to overcome Eisav’s angel — it is a mark of accomplishment. Pharoah at least hears this request but is not willing to give in. Moshe and Aharon realized the misunderstanding and immediately clarified. It was “Elokei ha’Ivrim,” that spoke to them — G-d of the entire tribe/family, the G-d of the “Ivrim,” the downtrodden slaves, not just G-d of the elite. Everyone needs to be let free to worship. This Pharoah is not even willing to hear.
It’s not “eved yisrael” — the term “yisrael,” as Pharoah understood, is one of chashivus. Rather, it’s “eved ivri” — a slave has no status. A slave has forfeited his membership in society, in the nation.