It is common nowadays to use "Golden Dawn" to refer to all the orders that share the bulk of the original Golden Dawn material, starting from the Cipher Manuscript "discovered" by William Wynn Westcott and most of it assembled or created by Samuel Liddell MacGregor Mathers.
We have books, blogs, web sites, tarot packs, and modern versions that blazon the name, including George Mills Haper's Yeats's Golden Dawn. However, during during Yeats's lifetime "Golden Dawn":
(1) was never used outside their own circle and was almost always presented as unexplained initials "G. D.";
(2) was properly applied only to the "outer" part of the order (the inner was the Order of the R.R. & A.C.—variously expanded as Rosae Rubeae et Aureae Crucis or Rubidae Rosae et Aureae Crucis, both meaning "red rose and golden cross");
(3) was dropped in 1902 following the scandalous Horos court case* and replaced with M.R. (the German Morgenröthe, "dawn", lit. "morning redness"); and
(4) disappeared when the renamed order dissolved over the next few years into several schismatic groups, including the Stella Matutina, the successor group that WBY was involved with into the 1920s.
Confusion can arise if all the successors are referred to casually as "Golden Dawn", so that Charles Williams's involvement with the Fellowship of the Rosy Cross can be conflated with Dion Fortune's involvement in the Alpha et Omega or George Yeats's in the Stella Matutina.
A fuller form of the name now widely used is the “Hermetic Order of the Golden Dawn,” but Nick Farrell, who has both scholarly authority and practical interest in the history of the order, maintains in his biography of MacGregor Mathers, King over the Water,† that:
the Golden Dawn was never called the Hermetic Order of the Golden Dawn (HOGD). That name was invented by Regardie for his book. The First Order was called the Golden Dawn in the Outer (GDO) although on some letterheads it was called the Esoteric Order of the Golden Dawn. Hermetic Order does appear on some letter heads and course material but it was never the Order's name.
(King over the Water, 17, [referring to this blog posting].)
Certainly the form of "Order of the G. D. in the Outer" is what appears on Yeats's invitation to his "admission"—where he was also told to "Ask for Mr Mathers & the Hermetic Students", and the "Hermetic Students" was the name that Yeats used for the group in his autobiographical writing (Au 575, CW3 453):
As Farrell indicates, course material kept by WBY and his uncle George Pollexfen contains ownership plates with “Hermetic Order of the G. D.,” often with the “G. D.” overwritten with “M. R.” (i.e., "Morgenröthe").¶
Click on the images for clearer versions (though still poor resolution—apologies).
In the end, however, it seems impossible to fight for too much accuracy over a name that was possibly even hazy to those who used it at the time—when we only ever use initials, they take over from the full name. Though it entails major caveats, less explanation is required and it is almost accurate to say that Georgie Hyde Lees joined the Golden Dawn on 24 July 1914, sponsored by W. B. Yeats (Saddlemyer, Becoming George, 66-68).
The name "Hermetic Order of the Golden Dawn" is probably a modern invention, but is succinct and has become standard. In a similar way Yeats's project for "Celtic mysteries" at the Castle of Heroes in Lough Key are often now referred to as the "Celtic Mystical Order", a term that had no currency at the time (see Collected Letters 2, 663-69), .
* For the Horos case, see chapter 1 of R. A. Gilbert's, The Golden Dawn Scrapbook: The Rise and Fall of a Magical Order (York Beach, ME: Samuel Weiser, 1997).
† Nick Farrell, King over the Water; Samuel Mathers and the Golden Dawn (Dublin: Kerubim Press, 201); see also Mathers' Last Secret (revised): The Rituals and Teachings of Alpha et Omega (Rosicrucian Order of the Golden Dawn, 2011).
¶ See items passim in groups NLI MSS 36,276–36,280, listed in National Library of Ireland's Collection List No. 60, “Occult Papers of W. B. Yeats”, compiled by Peter Kenny.