1/ Logres
All that is known for certain about this name, which in Welsh is Lloegr, is that it begins to be used for England in the 12th century by works such as the Welsh The Dream of Rhonabwy and Chrétien's Lancelot and Perceval. Before (?) that it occurs in Welsh collections called triads, although it must be remembered that the triads, like almost all Welsh poetry, may not be ancient because they survive in "late" manuscripts--12th century and after. For instance, in Enweu Ynys Brydein yv Hynn ("These are the names of the island of Britain") in MS. Peniarth 50, National Library of Wales, has a triad on the three realms of Britain: "Lloegyr a Chymry a'r Alban"--'Logres (England), Wales, and Scotland'.

Logres is the usual name for England in the 13th century Vulgate Cycle of prose romances. Some late medieval romances derive the name Logres from the name Locrinus, Brutus's son in the legendary history of Britain (in which Brutus, Aeneas's great-great-grandson, settles Britain). In Geoffrey of Monmouth, Welsh Lloegr appears Latinized as Loegria. But Geoffrey does not derive Loegria from the name of Brutus's son Locrinus, even though he derives the name of Wales (Latin Cumbria; Welsh Cymry) from Brutus's son Kamber (in History of the Kings of England VIII, 15). The name Logres is considered merely poetic, and in Chrétien's romances the flagrant disregard for geographical fact (e.g., there are no boundaries between Logres and places on the Continent, such as the Forest of Broceliande in France), seems to confirm this notion.

2/ Britain
Often called Great Britain to distinguish it from Brittany or Little Britain in France. Welsh Brydein or Prydein; Latin Britannia; West Germanic *Brituna > Old English Breoton, Breoton-land. Apparently, Old French Bretaigne and Middle English Bretayne derive from the Old English rather than from the Latin, as evidenced by the medial vowel. Old Celtic seems to have had no name for the island as distinct from the people, who were called Britto in Old Celtic and in Latin. After the Norman Conquest "Britain" referred only to the island in the legendary and historical past, until the time of Henry VIII and Edward VI, when it returned to political use as a neutral term for designating a united England and Scotland. In 1604 King James I was proclaimed King of Great Britain; the name "Britain" was adopted for the United Kingdom at the union in 1707.

3/ England
Old English Engla land. Originally the territory of the Angles, one of the tribes from modern Schleswig-Holstein in Germany who came to Britain as mercenaries at the request of Vortigern in the mid 5th century, and then conquered the island for themselves. Engla land distinguished land occupied by the Angles from land occupied by the Saxons. Oldest citation: ca. 890, in the translation of Bede's Ecclesiastical History of the English People (Latin, early 8th c) that is attributed to King Alfred. Later, "England" referred to the southern part of Britain, excluding Wales and the Danelaw (northern territory controlled by Viking invaders in the 9th and 10th centuries)--by this time the Angles and Saxons had become one group. After the Danelaw, "England" refers to Britain excluding Wales and Scotland.
 

Sources:
Oxford English Dictionary, 2nd edition (Oxford, 1989), s.v. England, Britain.

The Arthurian Encyclopedia, ed. Norris J. Lacy (NY, 1986), s.v. Logres.

G. D. West, An Index of Proper Names in French Arthurian Prose Romances, University of
        Toronto Romance Series, 35 (Toronto, 1978).

Charles and Ruth Moorman, Arthurian Dictionary (Jackson, Miss., 1978), s.v. Logres.

Trioedd Ynys Prydein: The Welsh Triads, ed. and trans. Rachel Bromwich (Cardiff, 1961).

J. S. P. Tatlock, The Legendary History of Britain (Berkeley, 1950), pp. 62, 149 (on Locrinus).