How many wives did Henry VIII actually have?

How many wives did Henry VIII actually have? According to “The Book of General Ignorance”, it was actually 2, or 4 if you’re Catholic…

Henry made himself head of the Church of England and then said his marriage to Catherine of Aragon was invalid.
Then, the Pope declared Henry’s marriage to Anne Boleyn illegal, because he was still married to Catherine of Aragon.
Later on, Henry annulled his marriage to Anne Boleyn, just before he had her executed. This is sort of confusing because as mentioned before, the marriage had not existed in the first place.
Henry’s marriage to Anne of Cleves was annulled. This is different from divorce, so means that a marriage never actually took place. He then showered her with gifts and called her his “sister”. She would visit court often swapping cooks, recipes and household gadgets (I think this bit was made up).
He then later annulled his marriage to Catherine Howard.
So, with Jane Seymour and Catherine Parr, that basically leaves us with two legal marriages.