This hadith has been reported on the authority of Anas through another chain of transmitters but there is no mention of the words (Muhammad, Messenger of Allah) in it.
Sahih Muslim Book 37, Hadith 94
← Previous Hadith Anas b. Malik reported that Allah's Apostle (ﷺ) had made for him a silver ring. and got engraved on it (Muhammad, Messenger of Allah) and said to the people I have got made a ring of silver and engraved in it (these words) (Muhammad, Messenger of Allah). So none should engrave these (words) like this engravement.

Sahih Muslim Book 37, Hadith 93 Next Hadith → Anas b. Malik reported that when Allah's Messenger (ﷺ) decided to write letters to the Byzantine (Emperor) they (his Companions) told him that they would not read a letter unless it is sealed. (Then) Allah's Messenger (ﷺ) had a silver ring made (for himself), (its shape is so vivid in my mind) as if I see its brightness in the band of Allah's Messenger (ﷺ) and its engravement was (Muhammad, Messenger of Allah).

Sahih Muslim Book 37, Hadith 95