Yes, I am in Kyushu. (The URL link I gave "says" the same!) This page shows 3 "Yufuin-no-Mori" trains (which is logical, since they go almost every hour). Now I can see that the two trains in my pictures are not the same. The second picture is taken from this blog; that train has golden stripes. From  what I can google, two (of the 3) trains are numbered as KIHA 71 and KIHA 72. (See also adjacent Wikipedia page.) A slide show with the third train is here.

On the ticket pricing: The distance spectrum appears to be approximately zonned. But as you notice from table prices, they do not give a single distance/price table. The prices are zonned from each individual station - usually, the same price holds for a few consecutive stops (of a local train). So there are fewer distinct prices than (near enough) destinations. I suppose that they adjust zonning for all directions from a single station simultaneously, so to need a "finite" total number of buttons at that station. I can imagine a bright newcommer, or a veteran JR emploee, solving this puzzle of regional price adjustment :-)

On internet, I found this example of a ticket vending machine (in Hiroshima):