From dictionary.com, "1780, from bath + room (n.). Originally a room with apparatus for bathing, used 20c. in U.S. as a euphemism for a lavatory and often noted as a word that confused British travelers. "
The word "Toilet" has a french origin.
for one thing your definition points out that it doesn't become a euphemism until the 20th century. The reason why it does then is explained by Pinker at the start of his book Blank Slate. Essentially it's just an example of the euphemistic treadmill (a term he coins). An example of this is the move from "idiot" to retarded to intellectual disability/handicapped.
switching to bathroom was a way of making the term for the room you take dumps in appear less crude; however, the nature of what you do in said room leads to this euphemism no longer being a euphemism.
looking at the oed it doesn't become a mainstream term until at least the late 19th century (and probably later).
if you are interested check out the first 30 pages or so of pinker's book as he goes into a good amount of detail about this specific move.