The trapezium article at MathWorld provides the following explanation (referencing OED) of how the U.S.- and U.K.-differing definitions arose:
According to the Oxford English Dictionary, the confusion of trapezium and trapezoid between the United States and Great Britain dates back to an error in Hutton's Mathematical Dictionary in 1795, the first work of its kind in the United States, which directly reversed the accepted meanings. Hutton assigned trapezium to "no sides parallel" and trapezoid to "two sides parallel" (Simpson and Weiner 1992, p. 2101).
After 1795 in the United States, the Hutton definitions became standard, while in the British empire, the Proclus definitions remained standard. Two hundred years later, the controversy remains. Country by country, region by region, and even teacher by teacher, the definitions of trapezoid and trapezium are commonly swapped.
Note, "the Proclus definitions" were mentioned earlier in the MathWorld article:
Proclus (also Heron and Posidonius) divided quadrilaterals into parallelograms and non-parallelograms. For the latter, Proclus assigned trapezium to "two sides parallel," and trapezoid to "no sides parallel." Archimedes also defined a trapezium as having precisely two parallel sides (Heath 1956, pp. 188-190).
Edit: Regarding distinctions made by Euclid, Proclus, Archimedes and others,
on pages 189 and 190 of volume I of Thomas L. Heath's The Thirteen Books of Euclid's Elements (1956) we find:
As Euclid has not yet defined parallel lines and does not anywhere define a parallelogram, he is not in a position to make the more elaborate classification of quadrilaterals attributed by Proclus to Posidonius and appearing also in Heron's Definitions. It may be shown by the following diagram, distinguishing seven species of quadrilaterals.
The diagram from Heath (representing quadrilateral classification per Posidonius, Proclus, and Heron) is shown below. One may compare this diagram with that shown in wikipedia's quadrilateral taxonomy section. That taxonomy does not have the word trapezoid in it, because it says "trapezium here is ... British" and "inclusive definitions are used throughout". (A British trapezoid is a convex quadrilateral that is not a British trapezium. The only categories of the taxonomy that can contain a convex British trapezoid are the quadrilateral, simple, convex, tangential, cyclic, kite, and bicentric categories.)
Heath then continues:
It will be observed that, while Euclid in the above definition classes as trapezia all quadrilaterals other than squares, oblongs, rhombi, and rhomboids, the word is in this classification restricted to quadrilaterals having two sides (only) parallel, and trapezoid is used to denote the rest. Euclid appears to have used trapezium in the restricted sense of a quadrilateral with two sides parallel in his book περι διαιρεσεων (on divisions of figures). Archimedes uses it in the same sense, but in one place describes it more precisely as a trapezium with its two sides parallel.
I think the first it in the last sentence refers to trapezium but don't understand what the second it refers to. Consequently, it is unclear to me whether MathWorld's assertions re Archimedes' definitions are correct.
Edit: In a
trapezium and trapezoid article on The Linguist List, we read the following quote from Oxford English Dictionary (under sense (c) of trapezium):
An irregular quadrilateral having neither pair of opposite sides parallel. (The usual sense in England from c1800 to c1875. Now rare. This sense is the one that is standard in the U.S., but in practice quadrilateral is used rather than trapezium.)
This indicates that during the 80 years after Hutton (a much-published English mathematician, born 1737 at Newcastle-on-Tyne, 1778 Copley Medal recipient) gave the definitions now in use in the U.S., those same definitions were also in use in England, afterwards falling into disuse (ie being reversed or set aright) in England. The OED goes on to imply that in the U.S., the all-encompassing term quadrilateral rather than (U.S.) trapezium is used to refer to quadrilaterals that do not fall into more-specific categories such as squares, rectangles, parallelograms, (British) trapezia, etc.