A fourth union of Elves and Men??

[Legolas]

Luthien was an elf. The point of Arwen and Dior isn't only that they are simply not entirely man or elf; it is that they are already products of man-elf unions. If you want to count the number of 100% man + 100% elven unions, your statement is correct, but that's not being discussed/disputed. The question is about the statements Tolkien made referring to unions within the stories that were made public (which means Mithrellas' case isn't included).

[Lachwen]

Elves and Men are evidently in biological terms one race, or they could not breed and produce fertile offspring

Oh dear, I hadn't thought of that. *promptly disappears in a puff of logic* Though I should have, seeing as how both my parents are scientists... Maybe it's a good thing I'll be starting school soon, my brain has obviously atrophied over the summer.

[Voralphion]

However, Tolkien's exact words were that there were three unions of the Eldar and the Edain. Now, the Eldar were originally the name of all Elves, but then only those who made the Great Journey to Valinor were known as Eldar afterwards

Actually the Eldar were all those who started the journey not just those who finished it. The only elves who were not counted as eldar were the avari. The teleri who never made it to Valinor were still Eldar, but they were Moriquendi and Calaquendi. Even the Nandor who turned from the journey before entering Beleriand were counted as Eldar.