In Hilchot Issurei Biah 13:17, the Rambam states that:
גר שלא בדקו אחריו או שלא הודיעוהו המצות ועונשן ומל וטבל בפני ג' הדיוטות ה"ז גר אפילו נודע שבשביל דבר הוא מתגייר הואיל ומל וטבל יצא מכלל העכו"ם וחוששין לו עד שיתבאר צדקותו ואפילו חזר ועבד כו"ם הרי הוא כישראל מומר שקידושיו קידושין ומצוה להחזיר אבידתו מאחר שטבל נעשה כישראל ולפיכך קיימו שמשון ושלמה נשותיהן ואע"פ שנגלה סודן:
Or, in English:
A convert whose intentions were not checked nor who was informed about the commandments and their punishment, but was circumcised and immersed in front of three laymen: he is a convert. even if we know that he is converting for an ulterior motive, if he was circumcised and immersed he is no longer considered a gentile, although we are suspicious of him until his righteousness becomes apparent. and even if he returns and worships an idol he is considered as a sinning Israelite.
Some ascribe this to mean that if the convert is later questioned, that he is a safek and needs a ger l'humra in order to be counted in a minyon, in order to get an aliyah, and so on. They treat a safek like a din where he is not "in the covenant." However, I prefer the historical explanation of the Rambam that follows the latter opinion you gave in your question. Rambam also goes on to explain in Hilchot Issurei Biah 14:2 that:
ומודיעין אותו עיקרי הדת שהוא ייחוד השם ואיסור עכו"ם ומאריכין בדבר הזה ומודיעין אותו מקצת מצות קלות ומקצת מצות חמורות ואין מאריכין בדבר זה ומודיעין אותו עון לקט שכחה ופיאה ומעשר שני ומודיעין אותו עונשן של מצות
Or in English:
And we inform him of the principles of the religion which are the oneness of God and the prohibition of idolatry, and we expand upon this. And we inform him of some easy commandments and of some hard commandments and we do not expand upon this, and we inform him of the sin of leket, shikheha, peah and maaser sheni [agricultural commandments], and we inform him the punishment of the commandments.
Notice that Rambam says "And we expand upon this," and "We do not expand upon this." One aspect to note about the Rambam in his Mishneh Torah is that not everything in it is a halacha. There are some non-halachic pieces of advice that he gives in this work, just as Pirkei Avot in Mishnah is fairly non-halachic as well.
I claim that the bold parts are his pieces of advice and not halachic in nature because we do not see them from the source text. Rambam is quoting Yevamot 47a, where it states:
Our Rabbis taught: If at the present time a man desires to become a proselyte, he is to be addressed as follows: 'What reason have you for desiring to become a proselyte; do you not know that Israel at the present time are persecuted and oppressed, despised, harassed and overcome by afflictions'? If he replies, 'I know and yet am unworthy', he is accepted forthwith, and is given instruction in some of the minor and some of the major commandments. He is informed of the sin [of the neglect of the commandments of] Gleanings, the Forgotten Sheaf, the Corner and the Poor Man's Tithe. He is also told of the punishment for the transgression of the commandments.
We see that gemara states that we inform of him of the minor and major commandments, that we inform him sins of the agricultural commandments, but we see nothing mentioned of expanding or not expanding on certain aspects of Judaism, or, as it appears, not even about the oneness of God or the prohibition of idolatry. Rambam, when he quotes a halacha, quotes it verbatim, or, at least in paraphrase, from that of Gemara. This is how Mishneh Torah typically works, which is why we see the depth and authority of his work even though he never directly cites any of his halacha (outside of maybe a verse from Torah).
As argued in this paper, the Rambam understood that the requirements for a valid conversion are largely hinged on ritualistic requirements. While it is the responsibility of the Beit Din to inform the convert and to make sure his intentions are sincere, the Beit Din, itself, does not get to decide whether they thought the convert was good enough. They serve as witnesses to his conversion, not as the gatekeepers. The paper argues that Rambam felt that, personally speaking, conversion should require devote faith, but that he understood that conversions themselves simply require what the halacha states, and that even a convert who "lied through his teeth" is still considered a sinning Israelite.