Tough call. I personally have an i5-2300k dual core processor in my machine, with an ATI 5670 graphics card, but Rift comes out in low 20's FPS at medium-high graphics settings (although I get low 50's in SWTOR, SC2, and Diablo 3 at High graphics settings, albeit with shadows low/off). In my experience I've experienced poor lifespan with AMD processors, they scale very poorly at higher temperatures, which is common with games using intense graphics processing. However I find better versatility with AMD/ATI graphics cards for multi-desktop viewing, especially since I tend to play with my game in a fullscreen window and have a web browser up on the other monitor (spoiler sites or You Tube playing most often). Still, I'd say processor over any other hardware, and go for rig #2 if you're set between those two choices.
Personally, I've shopped for my last two computers from cyberpowerpc.com after being recommended there by a friend, and I'm very happy with both the hardware and post-sales support I've received. Here's a PC I threw together with some similar specs (i5-3570k quad core processor, 2x4GB Crucial memory, ASUS P8Z77-V LX motherboard, dual AMD Radeon HD 6670 1GB video cards, etc.), price is $935, plus $75 shipping inside US, so you've got some wiggle room to add to the specs and still be within your budget. I hope you find something you like!
http://www.cyberpowerpc.com/saved/1E6MNA