To wear a coat like a cape (without inserting one’s arms into the sleeves) should preferably be avoided (see The Shabbos Home, pg. 107). According to Rav Moshe, this would be considered carrying.
ב"ה I do not have the book you mentioned in my possession so I can only comment that in the Shemirat Shabbat Kehichata, he also makes the reservation that it is preferable to refrain and in what is considered a complete Reshut Harabim "Reshut Harabim Gemura" which in most places, we don't have such a thing. However, as we shall see many did not make this reservation. As far as Rav Moshe Feinstein zt"l, if you have a specific source, I would like to see it . In any case, you are certainly entitled to be stringent, and there is no reason to ask me. On the other hand, the Shulchan Aruch 301:27, writes specifically that one may go out on Shabbat wearing a Tallit around him as long as it is not just hanging from his neck. Also in Shulchan Aruch 301:34 writes that in regard to a "sudar" which is similar to a scarf one may wear it even on his neck since that is "derech malbush", the way it is usually worn. Also, In the Responsa of Rav Efraim Greenblat zt"l, ((רבבות אפרים א: רכג: יד says it may be worn so and he consulted with other Rabbis on the issue who said it is permitted. He was also told by Rav Chaim Kanievsky that the Chazon Ish permitted to do so, and Rav Meir Greinman also wrote in his book that that the Chazon Ish permitted to do so. He also quotes the Be'er Moshe (Vol. 3:63) who also permitted to do so, though he says those who want to be stringent should be blessed. However, Rav Greenblat himself holds that there is no reason to be stringent. This is also the opinion of Rav ben Tzion Abba Shaul zt"l in (אור לציון תשובות ב: פרק כג: ז-ח) who permitted with no restrictions to wear a coat over the shoulders without putting arms into the sleeves. Furthermore, he permitted even to wear a towel over the shoulders for someone who went to the Mikveh on Shabbat. In short, what I wrote in my previous answer, was not my personal opinion, though one may deduct so from the aforementioned Shulchan Aruch. I thank you for the opportunity for looking up the matter a bit further which shows that this is the opinion of many Gedolei Yisrael. Nonetheless, I would very much appreciate if you could scan and send me a copy of what you read. All the best and Chag Kasher Ve'same'each