Sorry no one responded sooner. The fix is simple. If you look at the actual label sheet, you will see that it says to use Avery Template 5266. I created a label layout using the 5266 template, and the names print correctly on the 5866 sheets of labels. I have no idea why Avery changes numbers with multiple numbers for a particular template.
Often the difference in Avery number indicates the material from which the label is made--some are better suited for ink jet printers, others for laser, others are transparent... as well as the size of the label.
Of course those differences are meaningless in terms of how you set up software to print to them, so Avery usually includes "use Avery Template XXXX" info on the packaging to help users figure out what layout option to use.