To divorce (according to the Oxford English dictionary) is to end a marriage. In modern times usually by a formal legal process.
To annul a marriage is to say that the marriage never legally took place. There are assorted legal grounds for this in modern times, including bigamy and non-consummation.
Strictly speaking three of the marriages of Henry VIII were annulled and so never happened, so you are right. But the three wives of Henry VIII would not sound nearly so interesting.
Since he 'appeared' to have been married to the three in question, we would normally say that the annulments resulted in divorces, and you could use either word to describe the end of Henry VIII's marriages to three of his wives.
For those working on 'divorced, beheaded, died, divorced, beheaded, survived', note that Anne Boleyn's marriage was annulled shortly before she lost her head.
There's not a really satisfying answer to this, because the word "divorce" was in fact used from the beginning to refer to Henry's separation from Katherine of Aragon. When Thomas Cranmer was appointed Archbishop of Canterbury in 1532, for instance, Eustace Chapuys wrote that "He is a servant of [Anne Boleyn] and should be required to take a special oath promising not to meddle with the divorce. It is suspected that the new Archbishop may authorize the marriage [between Anne and Henry] in this Parliament."
In general, the widespread use of "divorced" likely reflects both the greater pop cultural presence of divorce than annulment, as well as the famous ditty, "divorced, beheaded, and died; divorced, beheaded, survived". It's beyond me to say when and where it was invented; the earliest I can find it referred to as a "doggerel couplet" is in a 1912 issue of Everybody's Magazine. However, it seems to have been fairly common for authors in the mid-to-late nineteenth century to present Henry VIII's wives in such a way as to suggest this kind of list.
The earliest I can find something like this is in the 1846 A Pictorial History of England, by Samuel Goodrich. It discusses the history of Henry's reign over the course of several chapters, and ends the one on his death with a list of his family, presenting his wives in a paragraph:
Catharine of Arragon, whom he divorced
Anne Boleyn, whom he beheaded
Jane Seymour, who died a natural death
Ann of Cleves, whom he divorced
Catharine Howard, whom he beheaded
Catharine Parr, who survived him
While this is not presented as a mnemonic like "divorced, beheaded, and died; divorced, beheaded, survived", it strongly suggests the repetitive chant. Similarly, The Mnemonic Chronology of British History, from the Roman Invasion to the Present Time (1849) reels off the women in a sentence that emphasizes the parallel structure:
Henry VIII, 1509, son of Henry VII, took the title of supreme head of the church, dissolved the religious foundations, and introduced the Reformation. He married Catherine of Arragon (divorced), Anne Boleyn (beheaded), Jane Seymour (died), Anne of Cleves (divorced), Catherine Howard (beheaded), and Catherine Parr, who survived him.
Later books presenting the entire history of England in brief would follow Goodrich's lead, to the point that the list was preserved, encapsulated in a box, as a standard table. The earliest version of this that I've found is in A Summary of English History by W. Reep (1882), and looks like this:
| Name | Children | Fate |
|---|
This table appears in a number of other publications from the end of the nineteenth century and beginning of the twentieth. It's easy to see how people would have simply turned that last column into a chant, or, potentially, how the chant would have suggested to writers to arrange the list of Henry VIII's wives in this way. But it is unfortunately impossible to say which came first.