PCPartPicker part list /
Price breakdown by merchant
CPU: Intel Pentium G3258 3.2GHz Dual-Core Processor ($69.29 @ Amazon)
Motherboard: Biostar B85MG Micro ATX LGA1150 Motherboard ($56.99 @ Amazon)
Memory: Kingston XMP Blu Red Series 8GB (2 x 4GB) DDR3-1600 Memory ($75.99 @ Amazon)
Storage: Seagate Barracuda 1TB 3.5" 7200RPM Internal Hard Drive ($51.98 @ OutletPC)
Video Card: Asus Radeon R7 260X 2GB DirectCU II Video Card ($114.99 @ Micro Center)
Case: Rosewill Galaxy-03 ATX Mid Tower Case ($29.99 @ Newegg)
Power Supply: Corsair Builder 430W 80+ Bronze Certified ATX Power Supply ($19.99 @ Newegg)
Optical Drive: Asus DRW-24B1ST/BLK/B/AS DVD/CD Writer ($19.98 @ OutletPC)
Total: $439.20
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2014-08-20 01:30 EDT-0400
This is the cheapest build I could possibly come up with, without sacrificing anything and including Windows.
The builds you provided have their pros and cons.
Ad 1 has a lot of potential. The graphics card will provide great performance as well as the fx 6300 processor being quite capable doing most anything, with a decent overclock on it (which may mean you need to upgrade the CPU cooler soon). I'm assuming the build comes with an OS which is a definite pro, as I cannot provide a copy of Windows without a 100 dollar surplus on the price. The RAM at 4 gb, won't be a huge problem except in battlefield 4, which is kind of demanding in multiplayer. The SSD is a definite perk as they add a lot of performance to bootups and access times to common files that you would store on the PC. The only problem with a SSD is the more you write to them the shorter their life gets. This being said, you will need another HDD, sooner rather than later. The graphics card and processor combo should allow you to play swtor on medium/high settings, as well as BF4 on low/medium settings. League of Legends doesn't require much to run.
Ad 2 provides water cooling, but on a 4300, that's pretty much a waste, even with an overclock you're going to struggle getting solid performance in games, causing a huge bottleneck. The large HDD is a nice perk as well as the 8 GB of RAM. The main down side to this build is the processor and the graphics card, if you could even consider a 6450 a graphics card with it's barely existent performance. The gaming keyboard and mouse are also added perks to this build. The motherboard and AIO liquid cooler does allow for future processor upgrades as well as support for new graphics cards, providing the 600w PSU is of decent quality. For this build, you will certainly need a better graphics card if you expect BF4 to even load. You will be able to play swtor, but most likely on the absolute lowest settings. League of Legends is a fairly lightweight game, so that is of no concern.
My build is quite a bit over what I assume is your budget, right at around 300-350 dollars. The main thing I'd like to point out about my build is, well everything. Everything is an upgrade path. The motherboard allows for use of all the newest processors in the haswell refresh lineup. The HDD is a quality 1 TB drive and for 50 bucks, you can't beat it. The graphics card is going to allow you to play at 1080p on just about high settings on BF4, as well as capping swtor and LoL. The PSU is no tier 1 power supply, but it will keep all of your hardware powered clean and efficiently. The processor is a dual core, but with a little overclocking, there should be plenty of leg room as these processor overclock extremely well.
All in all, if I had to choose a best case scenario for you, Ad 1 would be the best choice. I feel that it offers the best bang for your buck, and you'll only need to add a 50 dollar hard drive and possibly some RAM down the road. One word of advice, I have not personally dealt with AMDs coolers, but everyone always says they're junk, so you might want to invest in an aftermarket cooler as well. The Cooler Master Hyper 212 is a 25 dollar cooler right now, and it's got amazing performance. However, if you're able to convince the parents to spend a little more money on you, then the build I threw together will offer a nice upgrade path, and you could jump up to an i5, which is what most people rave over for gaming nowadays. (I quite enjoy my i7 =D) On top of that, you'd have all brand new parts with warranties and a little piece of mind if anything would happen to go wrong, rather than trying to chase some craiglister down because your rig died a year down the road.
I hope this helps, sorry for the lengthy post.
EDIT: Sorry I just got done proof reading my post and realized you said you had a monitor and OS. Fixed the build list.