I was torn between the same computers you are and I went with the x230t. Don't do the yoga, although its touchscreen and new it isn't one of the best out there. Also it depends on the yoga you are looking at. The only real one that can "compete" with the either the Twist or the x230t would be the Yoga 13. The Twist doesn't offer a digitizer pen where as the x230t does. The x230t and the Twist can run loads of low end games and some other ones such as Skyrim, Portal 2, and the new Tomb Raider. They can also run games like DOTA 2 and LOL. From what I can recall I believe the Twist may be slightly better in this regard but I'm not positive. The Yoga doesn't offer a lot of Hard Drive space, so if you need to save files or have a lot you need to keep, it would rule the Yoga out.
Yoga 13 specs
http://shop.lenovo.com/products/us/tech-specs/laptop/ideapad/yoga/yoga-13/
Another thing with the Yoga is that the battery is internal. Which I didn't care for because I feared that if my laptop battery died for whatever reason and I was no longer under warrenty I wouldn't have a way to replace it. Again memory was a BIG issue and selling point for me. Also keep in mind the way the Yoga flips the keyboard would be up against your arm or whatever surface you have it placed against.
Twist Specs
http://shop.lenovo.com/products/us/tech-specs/laptop/thinkpad/thinkpad-twist/
Ok, so I almost went with this computer and the only reason I didn't is because the x230t is better in a lot of regards. This computer is not digitizer pen friendly, you have to get another type of pen to use the screen as you would the x230t. This still has a lot of great features though, like the five point screen touch and good hard drive space. If you get this computer I would suggest just go all the way, (get the i7 and what not). It will last you as long as you treat it well. When you convert this laptop you don't have to worry about the keyboard being pressed down on like the Yoga.
X230t Specs
http://shop.lenovo.com/products/us/tech-specs/laptop/thinkpad/x-tablet-series/x230t/
Ok so here is why I picked the X230t. I wanted something that was going to last me a long time, not only in computer life span wise but in battery life too. This computer was made to handle windows 8 (although it doesn't come with it, keep that in mind). This also comes with a pen and you can get a docking station for this computer. Another selling point for me, was the keyboard which has a backlight. It is great for late nights when you can't see your keyboard (it has the standard two settings of backlight). You don't have to worry about your keyboard when you convert this one as well. Some other really nice things about this one is the 3G/4G ability. You can get a data card for this computer and use it just about anywhere. Also do not get the Outdoor display considering it adds a lot of glare and gloss to your screen.
A con to both the X230t and the Twist would be that the screen moves if you are typing too hard or not on a flat surface (like your lap).
Another note, whatever you do, DON'T GET SOMETHING WITH WINDOWS RT! You will regret it. It does not run things right and you can barely use your computer properly.
I hoped this helped :) I did about a month and a half of research before I picked the x230t. However if you have a bit more money you should consider the Helix. It just came out but as far as I know it has the best of all of these; you get the digitzer pen, a better screen resolution, and the ability to take it off the keyboard and use it as a tablet. Although it is considerably more (around $2000) or so.
Also another tip, look at videos on youtube and reviews of each of the products to help make your decision.
I got my X230t on ebay, and I would say you should consider getting your device there was well (make sure you get all the right specs and go for the "buy it now" option not bidding :) )
Edited for clarification