Both engagement rings and wedding rings have interesting histories and traditions.
Married women wearing rings to symbolize their married state goes back to at least ancient Rome, and wedding rings appear all over the world in many religions and cultures.
In the 1962 Rite of Marriage (which is word for word the same as the 1615 one published after Trent), you may notice that there is only a wedding ring given from the groom to the bride. Later editions of the rite add an option for a “double ring exchange/ceremony.” This seems to imply that the most traditional route (for Roman Catholics, at least) is only the wife having a wedding ring. However, scholars on the history of wedding celebrations, like The Nuptial Blessing by Kenneth Stevenson, reassure us that double wedding ring exchanges were quite commmon. (In Rome’s attempts to consolidate and standardize wedding customs, it is possible that some things were too obvious to write down.)
While diamond engagement rings as the norm are a completely manufactured custom spawned by the De Beers diamond monolopy in the twentieth century, engagement rings are a very old tradition. In the Christian West, one of the earliest references to an engagement ring is in a letter from Pope Nicholas I in 860.
Fielding, William J. Strange Customs Of Courtship And Marriage, 1942. http://archive.org/details/in.ernet.dli.2015.84585.
Jones, William. Finger-Ring Lore; Historical, Legendary, Anecdotal. London, Chatto and Windus, 1877. http://archive.org/details/fingerringlorehi00jonerich.
Maskell, Joseph. The Wedding-Ring: Its History, Literature, and the Superstitions Respecting It, 1868.