I see no one has answered you in the context of the scripture that you cited. Your Rabbi is right; but, see in scripture whenever it's speaking of the ger it means convert, but the foreigner can be a bnei Noach if he lives in the land of Israel, or just a foreigner that obeys the 7 just for the wisdom in them, but the later is no bnei Noach because he's not doing it to join himself to HaShem. They are only called the wise of the nations. The foreigner starts becoming a stranger when he seeks to do more and more mitzvos. A stranger is the one that has it in his mind to be a full citizen, were as the foreigner does not. Nor does he have the rights of a full citizen. He is under common law or universal law, were as the stranger is under Israeli national law. As shown in Exodus 12.43.
Then we see when it speaks of keeping pesach the stranger is mentioned to keep it but not the foreigner Exodus 20.10. You should also know that when the stranger is singled out; with words like don't abuse/mistreat them, it simultaneously also means the foreigner, evidence being the following words "you were strangers in the land of Egypt". Where the Israelites converts in Egypt? No, maybe some but not for the most part. All the do and don't do to the stranger commandments always apply to the foreigner too. You have to be a foreigner to get labeled stranger. But when it says foreigner that strictly means the one who is not a citizen, even if he is in the process of conversion.
Now Isaiah's sermon can be understood with this in mind. By saying foreigner he can address everyone and linking them with proper paralleling subject as we have it outlined in the Torah. Watch the clever thing he throws in there. Let not the foreigner say, who has attached himself to GD say "GD will keep me apart from his people." In the temple there was a separation wall. A court for the foreigners, how do we know here, it means stranger. Because the stranger is addressed alongside the eunuch, just as in Exodus 22.20-.21 but with widows and orphans. Orphans because according to HaLacha the convert has no parents. A woman convert addressed with the widow in the passage given above, because she may never find a husband, so Isaiah expounded on the teaching by speaking of the eunuch and the one in process to converting in the same sentence. It's an encouragement to them. And yes even the true Beni Noach must keep shabbat. I know you heard otherwise but only the gentile idol worshipers.
It says Takumim that keep the sabbath merit the death penalty. Why then do we make those taking conversion classes do something to not keep it perfectly. It's a hedge [Takanot] [enactment] to protect them. Remember he can change his mind at the very last minute about converting. Let's say he goes back to idol worshipping? That means an idolater has desecrated the Shabbat of HaShem. Its the reality of our world. Now I seriously doubt that bnei Noach ever kept shabbat in the manner of a Jew. He simply did not work his occupation. Notice the eunuch is keeping Shabbat, now we know that a man with crushed testicles can't become a part of the national congregation. Obviously it's a way of addressing the bnei Noach since the bnei Noach is like the eunuch in a small ways. That is that he has an issue in regards to his private area that keeps him out of the congregation. Obviously he's talking two different covenants because he's saying both that hold fast to my covenant but I just don't know enough to address the Shabbat keeping.