For some years there has been some debate on what the "true" name of the Golden Dawn was, whether it was Hermetic, Esoteric, Rosicrucian, or something else entirely. Some suggest it was one or the other, and that all other descriptors are wrong, but history says otherwise.
Firstly, if we look at the rituals themselves, the terminology used is "Order of the Golden Dawn in the Outer." This is, to the best of my knowledge, universal in all surviving copies of the original Order rituals.
The Ordinances give: "The First Order of the G.D. in the Outer." The Bye-laws of Isis-Urania give: "Order of the G.D. in the Outer." Circulars issued to members announcing meeting times also use this form. Likewise, the temple warrants use the same.
However, all labels on Order documents contained the title "Hermetic Order of the G.D."
To muddy the waters a little more, the title page of some documents gives: "Hermetic Students of the G.D."
Others say: "Hermetic Students of the Rosicrucian Order of the G.D."
In Westcott's "Historic Lecture to Neophytes," originally delivered in March 1888, he refers to it as simply the "Order of the G.D. in the Outer" (matching the rituals), but then calls it "an Hermetic Society."
Westcott further gives the name of the Order "in the several languages." These are:
"In Hebrew the title is 'Chabrath or Chevrah Zereh aour bokher' which means 'Society of the Shining Light of the Dawn.' [This is given as Chabrath Zerech Aur Boqer חברת זרח אור בקר in the Cipher MS.]
"While yet Latin was the language in almost universal use amongst persons of culture the name was 'Aurora'.
"In Greek Hē eōs Chrisē η εως χρυση.
"In French, 'L'aube Dorée'.
"In German the title is 'Die Goldene Dammerung.'"
In a response to a query about the group in the journal Notes and Queries, Westcott wrote: "Its true name is only told to initiates, and the few outsiders who have heard of its existence know of the society as 'The Hermetic Students of the G.D.'" He also referred to it by this name in a lecture to the SRIA.
A letter to Westcott in January 1888, purporting to be written by Anna Sprengel, calls the group simply "the Order of the G.D." In a letter to Yeats in 1900, Westcott refers to it as "the G.D. Hermetic Society."
A letter from Mathers to the Editor of the journal Lucifer referred to the Order as "The Hermetic Students of the Rosicrucian G.D. in the outer."
The initial Pledge Forms, meanwhile, gave the name as "the Esoteric Order of the G.D. in the Outer."
For those who rebelled against Mathers in 1900, the name was soon changed, largely in response to the very public Horos scandal of 1901. In the draft rules of reconstitution devised in 1902, it says:
"The name of the Hermetic Society of the G.D. shall be changed to some other title, to be approved by the Council."
The name chosen was Morgenrothe (Morning Red, or Dawn), and the letters "G.D." on most labels were replaced with "M.R." Bye-laws from this time also give: "Der Scheine des Lichtes" (The Shining of the Light) in brackets beneath "M.R. in the Outer."
In Mathers' post-rebellion group, the name was changed to "Alpha et Omega." The rituals describe it as the "Rosicrucian Order of A.O." or "Rosicrucian Order of the A.O." Surviving Bye-laws give "A.O. in the Outer."
However, the labels typically call it "Hermetic Order of the A.O."
Some papers, reproduced by photographic negatives, simply say "Order of A.'.O.'."
Stella Matutina labels appear to be much simpler, with the number of the temple. For example: "H.O. 21" (Amoun) or "H.O. 49" (Whare Ra/Smaragdum Thalasses).
As a matter of curiosity, one copy of a 0=0 ritual from the Amoun temple has "Stella Matutina" crossed out and replaced with "Monocris de Astris" (Unicorn of the Stars), suggesting this was a name being contemplated. It does not appear this was ever formally adopted.
So, clearly there was no single name that was universally used, and all of the above descriptors are correct in their own way. Perhaps the simplest name to use (in the case of the original Order) is "Order of the Golden Dawn," as this is largely consistent throughout the various forms, and the one adopted in both the official rules and rituals.