Actually I thought the seams on a basketball were two great circles and two ovals that each intersect the same great circle twice. And so apparently do many people who make basketball texture maps one can find on the web. (Randall Munroe too).
It is difficult to settle the question without having a physical ball to inspect; a usual photograph doesn't show both poles of the ball which is one needs. However, here's a photo showing that at least some basketballs are constructed like you describe:
I don't think the shape of the curved seam has a name -- judging from the photos I've found its precise shape can vary from ball to ball. The most realistic-looking images look like it's made up of four about 150° arcs of great circles, joined by curved segments near the poles, but how tight the joining curves are seems to vary.
From a construction point of view, the one-curve design does make more sense, because then the seams divide the surface of the ball into eight congruent pieces, each of which can be cut out of flat material without too much distortion (at compared to dividing into eights by three great circles), and also without needing eight-way meets at the poles. With two great circles and two ovals, the pieces would have two different shapes.
Incidentally, I don't think they're actually seams with modern manufacturing methods -- just a decoration that's put on the ball for reasons of tradition, and possibly to make it easier for players to perceive its spin.