A comprehensive history of SHAEF can be found at the US Army's Centre for Military History. The relevant volume of the US official history, The Supreme Command, can be downloaded here. Pages 275-78 have the history of SHAEF locations.
To summarise, at the time of D-Day the Main headquarters was in Bushy Park, near London, and the Forward HQ was at Portsmouth, both in the UK. The first advance HQ on the continent was opened at Tournieres, twelve miles southwest of Bayeux, on 7th August, and was quite small.
The Forward HQ started to move to Jullouville, France, on 28th August. This was near the HQs of the Allied Expeditionary Air Force, the Allied Naval Expeditionary Force, the U.S. Strategic Air Forces, and the French command, which were located in and around Granville. The Forward HQ started to move to Versailles on 15th September, and officially opened there on the 20th. A small advance headquarters was opened on 19 September at Gueux, about seven miles north-west of Reims, and the Forward HQ moved there on 17th February 1945.
At the end of September, the Main SHAEF HQ started to move to Versailles, and that was completed on 5th October. A Rear HQ was left in England, which moved from Bushy Park to Bryanston Square in London on 9-10 October.
So both the sources in the question are partly wrong, but an accurate answer is quite complicated.