Before I met my husband and moved to Hawaii, I, too, thought that rice was a matter of cooking it "right" to get a certain texture. Now I have a limited amount of storage space, but it still is normal for me to have three kinds of rice on hand.
If you want that lovely, separated, not very starchy feel to your rice, buy converted (Uncle Bens is the best known brand). Don't let anyone bully you into it "not really being rice". In fact, the method brought about to make it was meant to keep more nutrition in each grain, and it takes slightly LONGER too cook than others, and many restaurants use it for its pretty perfection. I use it for cold rice salads.
Basmati and Jasmine rices are different, but have similar uses. Most obviously Basmati for Indian food, Jasmine for Thai. Right now I have both on hand, but if I don't have one will use the other. They are VERY long grain rices, and can be delicate.
"rice cooker" rice ... CalRose is the best known brand, but short grain rice is what rice cookers were meant to cook. Not a separate grain, fluffly, un-starchy rice. This is a rice meant to be eaten with chopsticks. I use short grain rice interchangeabley, even have made (shock, shock, how dare you) risotto with it (arborio is a short grain rice). Definitely sushi. When I lived in Hawaii, everyone owned a rice cooker, and everyone used CalRose or similar, bought it by the 40 or 50 lb bag. Rice cookers were not meant to give you that look and feel of individual grains of non-starchy rice. They're meant to provide your family with a cheap-but-filling meal you can eat with chopsticks.
American long grain rice is somewhere in between all of these. It is something (Before Hawaii) was the only rice I knew. Now I rarely buy it, although it is a great product that spans everything and probably feeds a good portion of the world.
I just did a survey of my pantry for the heck of it and came up with some CalRose (short grain), Jasmine, Basmati (very long grain), a bit of converted (need more, salad season is here) and some brown (husband was on a whole-grain kick for awhile there, but suspect I'll never cook it and donate it to the food pantry, because he doesn't like it AT ALL, in spite of the whole grain kick).
OH! Wild Rice. Takes longer to cook, and is great with game. Chewy texture. Some say it isn't really rice, it's .... blablabla. Comes often in packages that combine it with long grain rice, but when the long grain is done, the wild rice isn't. One young Hawaiian acquaintance asked me, "why do they put sticks in their rice on the Mainland?" It is good, but cook it separately if you're mixing it with other kinds of rice, then mix.