What types of rice are appropriate to use in a rice cooker?
In my experience, rice quality has a pretty substantial amount of variation, but the cooking device has little to do with it.

While I'm sure that most rice cookers on the market from Japanese firms are optimized and tested for short-grain, japonica rice, I've never had issues cooking basmati or jasmine rice in one, and I've even used them for farro and mixed grains.

I doubt that the rice cooker has much to do with it. I've made rice in heavy cast-iron enamelware on the stove, in a cheap Southeast Asian electric steamer without any fancy "fuzzy logic" electronics, in a fuzzy logic rice cooker, and an mid-range IH rice cooker, which is what we use at home now, and the quality of the rice and its age has a greater impact than the cooking method. I would say that the IH rice cooker produces superior results over our old fuzzy logic cooker, but it's certainly not an order-of-magnitude kind of difference.

Assuming you're located in the US, my benchmark go-to rice brand is "Tamaki Gold", which is from a japonica strain called koshihikari, and I think most of this brand's rice is grown near Sacramento, CA. It's more expensive than the typical Botan or Niko Niko brand calrose rice that's ubiquitious in Japanese supermarkets, but I find the quality far superior, and it's still a good value. My wife tends not to appreciate the Niko Niko or Botan calrose rice very much at all, so we don't eat it at home, but probably 80% of Japanese restaurants in the US are using it or a similar product.

Generally, I'd recommend staying away from the absolute cheapest brands, and choose something that's a couple of notches above. We've used imported rice from Akita or wherever and gotten very nice results, but the differences were far subtler than the price (on the order of $10 vs. $35 for a similar quantity). The sweet spot for quality is near the median price, assuming you're in a shop that offers a wide variety of options.

Edit: Sorry, until your edit, I didn't realize you were working from parboiled or converted rice, I assumed raw rice, as I've never heard of anyone cooking the quick cooking rice products in a rice cooker. (Rice cookers often have their own "quick mode" which shortens cooking time with normal raw rice at the cost of a slightly reduced textural quality). Considering that's what you were starting from, consider using other supermarket brands of rice that meet your target grain size and stickiness/fluffiness, but aren't marketed for speed or convenience.