There's no doubt that both the Yeomen of the Queen's Body Guard (in-ordinary), whose headquarters is at St James's Palace, and the Yeomen of the Queen's Body Guard (extraordinary), who are the Tower Warders, are both known as Beefeater, like it or not! However, the nickname originated with the Yeomen of the Guard from St James's Palace, possibly because of their daily rations. See Rations

Evidence Part One: 

A story which would have been crucial in dating the term Beefeater, had it not been found to have been adulterated to suit the narrative and therefore distorted the research, is the story of Henry VIII (born 1491 – 1547), dressing as a yeomen of the guard and invited to dine with the Abbot of Reading. The story was related by a Dr Thomas Fuller (19 June 1608 - 16 August 1661) an English churchman and historian who wrote with authority that it was true. This story is taken from his book 'The Church History of Britain’, Vol II, Book VI, Section II, page 192, reprinted 1837:

‘King Henry VIII, happening to be hunting in the neighbourhood of Reading Abbey, disguised himself in the uniform of one of his Yeomen of the Guard, and for a frolic paid a visit to the Abbot of that place about dinner time; the Abbot receiving his guest, as one of the Royal retinue, with great civility, and invited him to dine at his own table, the principal dish being a large piece of beef, of which the King, hungry from the chase, ate voraciously. Observing this the Abbot cried out “Well fare thy heart! And here in a cup of sack I remember the Health of his Grace your Master: I would give a hundred pounds on the condition I could feed so heartily on beef as you do. Alas! My weak and squeamish stomach will hardly digest a piece of small rabbit or chicken!” The King (or rather the Beefeater) then departed, and in a few weeks afterwards the Abbot was committed close prisoner to the Tower of London, and fed for a short time on bread and water, at length a piece of beef was placed before him, which the Abbot attacked with right good will, while he was thus engaged, the King himself entered the apartment and demanded the £100 for having restored the Abbot to his lost appetite for roast-beef.’ Unquote. Thereafter whenever the Abbot saw a Yeoman of the Guard he thought of the beefeater, and the King in disguise as a Yeoman of the Guard.'

The tale is told by other historians with some slight variations, and it is just possible that the jocular name of Beefeater was given to the Guard when this story was repeated, as indeed it would be.

While attempting to corroborate  the origin of this story I found the same story quoted by Thomas Smith in 1852 who refers Dr Fuller’s story in a somewhat changed version.

‘King Henry VIII, happening to be hunting in the neighbourhood of Reading Abbey, disguised himself in the uniform of one of his Yeomen of the Guard, and for a frolic paid a visit to the Abbot of that place about dinner time; the Abbot receiving his guest, as one of  the Royal retinue, with great civility, and invited him to dine at his own table, the principal dish being a large piece of beef, of which the King, hungry from the chase, ate voraciously. Observing this the Abbot cried out “Well fare thy heart! And here in a cup of sack I remember the Health of his Grace your Master: I would give a hundred pounds on the condition I could feed so heartily on beef as you do. Alas! My weak and squeamish stomach will hardly digest a piece of small rabbit or chicken!” The King (or rather the Beefeater) then departed, and in a few weeks afterwards the Abbot was committed close prisoner to the Tower of London, and fed for a short time on bread and water, at length a piece of beef was placed before him, which the Abbot attacked with right good will, while he was thus engaged, the King himself entered the apartment and demanded the £100 for having restored the Abbot to his lost appetite for roast-beef’. Thereafter whenever the Abbot saw a Yeoman of the Guard he thought of the Beefeater, and the King in disguise'

Nowhere in Fuller’s version was the word Beefeater / Beef-eater used. If it had, the origin of the term Beefeater could have been dated as being used before Henry VIII’s death in 1547 and therefore would have definitely been attributed to the Yeomen of the Guard from St James’s Palace (in-ordinary), because the Tower Warders were not sworn as Yeomen of the Guard (extraordinary) until 1550, three years later. 

Evidence Part Two:

Certainly during Queen Elizabeth's reign her Body Guard were known as 'Beefs' as it shown in the 1571 and 1572 confessions respectively when Edmund Mather and Kenelm Berney were involved in the plotting to overthrow and indeed assassinate Queen Elizabeth. Mather’s confessions are below but in both he describes the Body Guard as ‘Beefs’. Both Mather and Berney were executed on 13 February 1572. In 'The History of the Reign of Edward the Sixth, Mary and Elizabeth Vol 2, pg 266'. it states:

‘….Regicide in some aspect presented the most temptation. " To kill a sovereign would make their fame immortal". “The Queen's Beefs” were poor creatures, a handful of determined men could easily dispatch, and the rest of the household were, "perfumed minions” such as the vile woman kept about her to feed her fantasy.’

‘….In his first conversations with Berny, Mather said, ‘If the court were the town, I would undertake to take the Queen and all her true men, with two hundred gentlemen; aye, with half the number’. This talk he used many times with me. He said the guard, calling them the Queen’s beefs, might soon be dispatched.’

Notwithstanding this earlier dated evidence I sought to investigate the alleged origin of the term ‘Beefeater’ that is most regularly quoted. One involving Count Cosmo III when he visited England in 1669. In Thomas Preston’s book 'The Yeomen of the King's Body Guard 1485-1885' states:   

‘The earliest credible documented reference of the nickname 'Beefeaters’, appears to have its origin in 1669, when Count Cosmo, Grand Duke of Tuscany, was in England, and, writing of the size and stature of this magnificent guard, said, 'They are great eaters of beef, of which a very large ration is given them daily at the Court, and they might be called beefeaters.’   

The Count’s visit is recorded in a book called ‘Travel of Cosmo the Third Grand Duke of Tuscany Through England, During the Reign of King Charles The Second (1669)’ written by his friend and companion on the visit Count Lorenzo Magalotti. He relates the story of the ‘beef-eater’ thus (page 308):

‘….employed to mount guard at the gates of the palace, both on the side of St James’s Park and that of Whitehall Palace, and to escort his majesty whenever he goes out on horseback or in his carriage through the city. In the hall called the Guard-Room, is the guard of the Manica or Sleeve (yeomen of the guard) consisting of two hundred and fifty very handsome men, the tallest and strongest that can be found in England; they are called, in jest, Beef-eaters, that is, eaters of beef, of which a considerable portion is allowed them by the court every day.’ 

In Count Magalotti’s account it’s uncertain if the nickname Beef-eater was coined by Count Cosmo or already in existence given that the text states‘…they are called, in jest, Beef-eaters, that is……’ The account is written as though the term beef-eater is already in general parlance.   However, it’s hardly surprising that the Count was so taken with the Yeomen at St James’s Palace and their rations when you see their daily table.