I've never used one, but a tile saw would enable you to cut cups in half and to cut plates, etc..I have a Taurus 3 ring saw and there is a blade you can buy which enables you to cut plates but you are limited in the height of what you are cutting. It cuts slower than a tile saw would. You can also use your ring saw to cut glass. It might not be a tool that you use every day, but for the times you do need it, you will be glad you have one. As far as cutting shapes, a Beetle Bits or Morton would allow you to cut strips, squares, rectangles, diamonds, etc...I think there are also circle cutting accessories you can purchase too?
Hope someone else chimes in which their opinions!

I have a Taurus 3 ring saw and I know that you can detach the saw from the stand and use it to cut larger objects that won't fit otherwise. But I never got up the nerve to use it that way! My friend has a tool that cuts circles. It works with a suction cup and an arm that turns around and it scores the glass.

Here's another good instruction article, from Enchanted Rose Studio.

Here is a link that might be useful: How to nip

I like cutting by hand. I dont use any power tools, I enjoy that, but thats me. To help with your hand cutting read up on stained glass, older books where they teach the art of cutting, not just whipping out a saw. I learned a lot that way.
I would enjoy having a tile saw at times for particular things, but like you, I enjoy the imperfections.

I have been experimenting with all the different cutting tools & have come to the conclusion that I need to buy me a ring saw. I guess I will be selling a lot of my scrapbooking supplies to pay for my new hobby lol. I have also purchased a dremel as a lot of people have had good luck with those as well. No matter what you use,if you are happy with the result then thats all that counts & in any hand made craft, doesn't have to be perfect :)
I have a tumbler that I tumble a lot of my cut china & glass with a little sand out of our back yard as we live on a sandhill, works great. Just 1.5 tablespoons per a 3 lb drum takes all the sharp edges off after about 30 minutes for china or pottery & a little longer for glass