What flight said, and for the low-end CPUs AMD usually gives wattage overhead so that you can overclock decently even on the stock cooler.
Check out the updated build in my latest post, it's ~280$ and features both a more powerful CPU and a much better GPU, albeit integrated . I can't emphasise on this enough - do not go for the GT 610/710/720 cards. You will be massively disappointed by their performance. At your budget, an APU, or otherwise said a CPU with a GPU die on it, is the best route.
Also, if you're going to go with an integrated GPU, definitely get 2x4 GB of RAM instead of 1x8 GB. With two sticks, it becomes dual-channel and generally a lot faster, 64 bit bus instead of 32 bit. It might be 10$ more but it's ABSOLUTELY worth it, you will get as much as 20-30 FPS more in some games.
P.S. actually, on your latest list, the integrated GPU in the processor is more powerful than the discrete GPU. That should give you an idea of what the GT 610/710 actually is.