Considering that Obito, Kakashi and Madara all knew the name of Kamui without someone telling them, can we conclude that those who have the Mangekyō Sharingan know the name of its exclusive power? (Obito's Kamui, Shisui's Kotoamatsukami, Itachi's Amaterasu and Tsukuyomi, Sasuke's Amaterasu and Blaze Release: Kagutsuchi, and everyone's Susanoo)
Yes, only the ones with the Mangekyō Sharingan know the names of their techniques.
Taking an example, Obito and Kakashi had Obito's Mangekyō Sharingan. And they had the same Mangekyō technique (Kamui). Considering that Kakashi and Obito never saw each other or trained together after they awakened the Mangekyō, it is not a coincidence that both call their technique "Kamui".
In raw terms (sorry for the bad example), consider the Mangekyō Sharingan as a system, and it has two processes: A.exe and B.exe (the Mangekyō techniques). The two processes are already named, installed and stored in the system. And for the user to access it, all it needs is the fulfilment of the system requirements (loss of a loved one). So, when the uses has access to the processes, they naturally will know what to call it, since its already named and stored.
I hope this answers how the users know the names of their MS techniques. Nothing can be said about how the MS techniques are named because the current level of evidence is inconclusive.