EXACTLY what I said to them both, but they were having none of it

Bizarre. I mean, I get that it is a SkyUI convention that an arrow points to the selected category. But if maintaining convention is the worry, I'd still suggest that there are better solutions. You guys could move the arrow, put it centered above or below the selected tab (with the text perhaps centered under it). Or get rid of the arrow, and rely on the text highlight (and/or a background gradient similar to the highlighted list item) to indicate which list is active. Even adding a second arrow to the outer side of the tab, so there would be arrows pointing inward on both the left and right sides of the selected tab, might help. The positioning in these screenshots just seems needlessly confusing, with the two container labels so close together, and the single arrow between them, sized and positioned exactly as if to show the direction of transfer.
Granted it's not a big deal, I'm sure I could get used to whatever. I got used to the default interface, after all. But it just seems to me that there are fairly easy ways to have something more intuitive than this.