[NOTE: This is in response to a query on LiveJournal in which the submitter, The Dragon Demands, queried GRRM on details in recent episodes of the TV show Game of Thrones]
I suspect that "Maegor III" was a mistake, though I cannot say for certain. Perhaps a flubbed line, as you suggest. It is true that the Targaryen succession on the series is different than the one in the novels; most notably, the Mad King's father Jaehaerys II was dropped, as was established way back in season one. In much the same way as the Rhoynar have been dropped from the royal titles, "King of Andals and the Rhoynar and the First Men," etc.
These changes were simplifications, however. The books are very complex, but the practical limits of a television series call for a bit more simplicity. Dropping a king or two accomplishes that.
ADDING kings, however, would be a step in the opposite direction, which is why I think "Maegor III" had to be a mistake. And not one that was in the scripts, I would guess. Bryan Cogman, who is the Keeper of the Continuity on the series, knows the names of the Targaryen kings as well as I do.
Of course, it could also be a subtle bit of characterization, as you suggest, intended to show that Mace is an idiot who does not know his Westerosi history. (Not a mistake that Book Mace would make, but the character in the show combines Mace with Harys Swyft, and actually seems more like the latter).
All this, of course, is surmise on my part. You would have to ask David or Dan or Bryan for a more definitive answer.
In the book canon, of course, there has only been only King Maegor, the reputation of Maegor the Cruel being so black. England has had only one King John, for much the same reasons. (Prince Aerion Brightflame did name his son Maegor, but that was meant as a provocation, and in any case the boy never sat the Iron Throne).
As for the Night's King (the form I prefer), in the books he is a legendary figure, akin to Lann the Clever and Brandon the Builder, and no more likely to have survived to the present day than they have.