Rav Miller notes that throughout Tanach, Am Yisrael is called "Yisrael."
The surrounding nations tend to use the term Ivri/Hebrew...but Hashem & Am Yisrael tend to use terms like "Yisrael/Israelite," "Am Yisrael/Nation of Israel," "Bnei Yisrael/Children of Israel," and so on.
Rav Miller explains that when speaking to others in Tanach, Am Yisrael used the word Ivri because that's what the other understood.
(Sort of like how we say the English "God" instead of our more meaningful Hebrew or Yiddish terms because if you start going on about "the Aibeshter" or "Hashem," how many people will understand you?)
So Moshe Rabbeinu & Yonah HaNavi used the term "Ivri" when speaking to Pharaoh or the non-Jewish sailors.
But it's not a term we normally use.
Based on this idea, it's interesting that even today, Bnei Yisrael of ancient times tend to be called "ancient Hebrews."
That term makes us sound like a race, rather than a nation on a special mission.
Sure, that may have made sense until a certain point. But for the vast majority of Tanach, we are called "Yisrael."
Yet Rav Miller explains "Ivri" as a dismissive term. It literally means "passer-over"—someone who passed, like passed over the Euphrates.
(That's what it means for those who don't know. Spiritually speaking, Avraham Avinu passed from the profane to the holy. But that's not how the other nations mean it.)
It implies that the descendants of Yaakov Avinu are merely another branch of the Semitic tribe (from Shem)—they came from "over there"—in other words, nothing special.
Rav Miller also notes that the language Ivrit/Hebrew was generally called Lashon Hakodesh (The Holy Language) among Jews, and not Ivrit/Hebrew.
Here's another interesting point on page 6 that I never considered (but always found the name awkward & odd—now I know why!):
So you know now that the Union of American Hebrew Congregations is not a Torah organization. And if you read in the New York Times – you shouldn’t, but if you did – that a group of reptiles, of loathsome animals who marry a certain way, organized a congregation, a synagogue, and they received recognition from the Union of Hebrew Congregations, so we say,
“Of course Hebrews would do such a thing; but Yisroelim or Yehudim could not.”
Be Yourself—Your REAL Self!
Yisroel is a chauvinistic name, a badge of honor that we wear with pride, and therefore it’s understandable that the nations of the world are not interested in hearing such things.
This attitude ("We're nice & harmless! We're not so different than you!") has spread to parts of the frum community, especially among those frummies prominent among the general American population.
Clothing styles, speech patterns, slang, and even in foul language in some cases increasingly imprints itself on many frum people.
(For instance, it's weird to hear a born-and-raised Monsey FFB speak like a Valley Girl or a guitarist from California.)
Needless to say, we should be nice. We should behave with supreme derech eretz.
And even when the plain halacha doesn't obligate us a certain way, we should still take the high road.
For example, Rav Miller notes that the laws against lashon hara only apply to fellow Jews.
However, slandering & rumor-mongering stand out as mean behavior that makes you a mean & nasty person in the end.
Thus, even though you're not breaking the Torah law against lashon hara by speaking negatively about, say, your non-Jewish colleague...you ARE engaging in ugly behavior that ruins your character.
(https://torasavigdor.org/rav-avigdor-miller-on-gentile-lashon-horah/)
So you should not do it.
Our job is to rectify the world.
When you make a bracha over a glass of water, you are rectifying something in the world.
When you light Shabbat candles, you are rectifying something in the world.
When you put on tefillin, you are rectifying something in the world.
While doing all those rectifications, we should also lead the way in proper behavior.
We should look & act like bnei haMelech!
We should be very good!
But we aren't like everyone else, nor should we try to be.
With all today's emphasis on "being yourself!" and "be true to yourself!" (and hearing this said constantly in the frum community too), it's odd that being true to your Jewish neshamah is less emphasized.
After all, your neshamah is the REAL YOU!
The Most Important Function of a Jew
And it has nothing to do with being a jerk.
On page 10, Rav Miller briefly addresses the idea behind the more modest bat mitzvah celebrations for girls.
So the big tragedy for the eved Ivri is that he chooses to loose his special status as a Yisrael.
Let's end with a thought from Rav Miller (page 6):
And even after the destruction of the first Beis Hamikdosh when we lost the ten tribes of Yisroel and only Yehuda remained, so we took the name Yehudim – that’s also a name of glory because Yehuda means, “the one who praises Hashem.”
That’s the most important function of a Jew, a Judah, in the world; to speak about Hakodosh Boruch Hu.
We are the nation that speaks always about Hakodosh Boruch Hu.