Derek Cohen (in Perth, Australia)
Bill, the saw you link to is not the same as the saw in the photo. The saw in the link is a kataba, and does not have a back.
I would prefer to use a dozuki, as in the photo, which has a back. Just like a Western backsaw, the back offers a stiffer and therefore more stable cut. With dividers, you are not sawing deeper than about an inch, so depth is not an issue here.
Secondly, the teeth of most dozukis are very fine, somewhere in the range of 18 - 26 teeth. More importantly, the plate is very fine and the set is minimal. I have used rip- and crosscut saws interchangeably. I do not think this makes much difference.
If you plan to get another saw, look for the one in the photo, or a Z-saw. They are both cheap and very reliable. Rip and about 20 teeth. I use one for crosscutting the dovetail ends of drawer sides, or for precision cuts in thin stock, where the finish is important.
Regards from Munich (visited an amazing museum of design today - more later)