The First Fleet had arrived in this country over the three days from 18th to 20th January. It did not take long for Arthur Phillip to decide that Botany Bay was unsuitable for the settlement. After investigation, he chose Sydney Cove in Port Jackson instead.
To speed things up, Phillip decided to take the Supply up to Port Jackson first, leaving Hunter to follow with the rest of the Fleet. On January 25th, a strong flood tide and contrary winds impeded several attempts by the Supply to leave the Bay but eventually, it managed to clear the heads and turned north accompanied by two long boats.
Philip Gidley King, who was on board the Supply, relates that it anchored in Sydney Cove at about 7pm and that
The next day [the 26th] at Day light the English colours were displayed on shore & possession was taken for His Majesty whose health, with the Queens, Prince of Wales & Success to the Colony was drank, a feu de joie was fired by the party of Marines & ye whole gave 3 Cheers which was returned by the Supply.
At Sun sett The Sirius & all the Convoy anchored here.
For those aboard the ships of the convoy, the on-shore ceremony held that morning before their arrival had no special significance. They were probably unaware of it. Captain John Hunter, who had led the remaining ten ships of the convoy north from Botany Bay to Port Jackson, certainly makes no reference to it. His record of that day is more focused on his encounter with the French ships of La Perouse.
Nor was this brief morning ceremony mentioned in the "Voyage of Governor Phillip to Botany Bay" which was said to have been compiled from official records and published in London in 1789. It does mention an equivalent event that is says occurred on the evening of the 26th but in recounting this, it seems to be describing PG King's day-break ceremony. It is likely this publication just got the timing wrong.
There was however another ceremony held 12 days later that is given much more prominence in this "Phillip" publication and is also mentioned in the journals of several others.
The 7th of February, 1788, was a memorable day which established a regular form of Government on the coast of New South Wales. For obvious reasons, all possible solemnity was given to the proceeding necessary on this occasion. On a space previously cleared, the whole colony was assembled; the military drawn up, and under arms; the convicts stationed apart; and near the person of the Governor, those who were to hold the principal offices under him. The Royal Commission was then read by Mr D Collins, the Judge Advocate. By this instrument Arthur Phillip was constituted and appointed Captain General and Governor in Chief in and over the territory, called New South Wales ...
What followed was long list of latitudes and longitudes carefully describing what was being claimed as lying within New South Wales. Much of what we call Australia today was not included. This was to avoid infringing on what might be claimed by other countries based on their earlier "discoveries".
After the proclamation was read, Governor Phillip gave a speech "which was received with universal acclamations". Later ...
he gave a dinner to the officers, and the first evening of his government was concluded propitiously, in good order and innocent festivity, amidst the repetition of wishes for its prosperity.
And how does the "Voyage of Governor Phillip to Botany Bay" go on to describe the original inhabitants' reaction? It notes that attempts to have friendly discourse with them had been frustrated; they had been fired at and threatened; they had become "shy" of encounters with the Europeans and withdrawn into the woods; and that
Their dislike of the Europeans is probably increased by discovering that they intend to remain among them, and that they interfere with them in some of their best fishing places, which doubtless are, in their circumstances, objects of very great importance. Some of the convicts who have straggled into the woods have been killed, and others dangerously wounded by the natives, but there is great reason to suppose that in these cases the convicts have usually been the aggressors.
For David Collins, the February 7th ceremony gave legitimacy to the British claim to the land:
Great Britain alone has followed up the discoveries she had made in this country, by at once establishing in it a regular colony and civil government.
While January 26th gets all the recognition, it was from February 7th, 1788 that British rule of the land they called New South Wales officially began.