According to Seductions of Rice (which also has the best written instructions for making sushi rice I've seen), any Japanese-style rice will work fine for sushi rice. They further define Japanese Rice as Japonica short rice which has a length:width ratio of 2.5:1. The grains should look translucent and rounded, sometimes with a small white spot at one end.
For example, I personally use "Akita Komachi" organic rice. This is also a "half-brown" rice, which gives it more vitamin content than pure white rice, as well as a nice nutty flavor.
According to them, Japanese rice is frequently covered in talc or powdered starch to keep it dry, which is why rinsing it prior to cooking is essential.
I'm not going to go through their entire instructions for preparing sushi rice -- you can borrow the book from your local library for that -- but I will go over the essentials to make sure you're not missing major steps:
- Rinse the rice
- Soak the rice in cold water for 20 minutes
- Cook the rice
- Mix the rice with vinegar, sugar and salt
- Spread the rice out in a wide pan to cool
- Make sushi within 3 hours.
The basic differences between white and brown rice is explained below (as written here):
An important first question to ask about all rice-and for that matter, most foods-is how much it has been processed. In the case of rice, processing usually involves milling and polishing. The outermost layer of rice, called the hull, is removed to make brown rice. Brown rice is rice with the whole kernel intact and the kernel is still surrounded by all layers of bran.
To produce white rice, the bran layers of the rice have to be milled off. Most of the rice germ is also removed during this abrasive grinding process. At this point in the process the rice is called milled, unpolished white rice. Finally, a wire brush machine is used to remove the aleurone layer that remains on the rice. This step is called polishing. As polishing is not an all-or-nothing process, semi-polished rice may still contain parts of its aleurone layer.
Additional information about brown rice (here):
[Brown rice] has a mild nutty flavor, and is chewier and more nutritious than white rice, but goes rancid more quickly because the germ—which is removed to make white rice—contains fats that can spoil. Any rice, including long-grain, short-grain, or sticky rice, may be eaten as brown rice.
So, because they are processed differently and have different parts of the grain intact or removed, your method of cooking is going to vary.
I have used this method of cooking brown basmati rice with pretty good success. It includes washing the rice a number of times prior to cooking and adding salt. This method advises adding oil after the cooking process, but I typically add it into my rice while it cooks, or even sauteeing the grains in oil then adding water to the pot to cook.
It is also suggested that leaving your brown rice to soak for 15-20 minutes (and sometimes even longer) can be beneficial as well.
All of that aside: if you're looking to stick with a flavor and texture similar to that of white rice then brown rice may not be your thing. I know that as a jasmine rice eater I do not particularly care for the texture of brown (even brown jasmine). As mentioned above it does have a "nutty" flavor and is "chewier" regardless of how well/properly it is prepared. It changes the flavor and feel of every dish I serve it with. Just a heads up.
Best Answer
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.