What pc would be better for gaming first person shooters, minecraft and generally pc games?

I can safely tell you that those PC's aren't for gaming, the PC specialist one have a notebook GPU, which isn't optimized/compatible with desktops, I seriously don't know why they add a notebook graphics card to a desktop, anyway, the CPU is fine, the RAM is great, the HDD is kinda good, and the OS isn't good.
There are few games that actually like Windows 8.1. When I updated to Windows 8.1, my performance on all titles was trash, I had like 60+FPS on BF4 and with 8.1 it came down to 30FPS, so I wouldn't get this PC if you want really good performance.

The FX-4170 it's actually kinda good, it has a 4-core, 4.20 Ghz CPU, 8 gigs of RAM, 1TB of HDD capacity and a 6670 which, again, isn't good for gaming.

That GPU won't play 2013-2014 games at medium, which isn't good but is it better than the 8570D? Definitely. You could do this: You can buy the FX-4170 one and then if you can't play games smoothly, replace the 6670 with a more powerful GPU like a 7850 or a 7870, those would give you excellent performance, or you could also build one, it is fairly easy and it can be more powerful than these 2 and it could cost less

The A8-6600K is NOT a laptop CPU. But, you are referring to systems at 2 different prices, and the one at Amazon is better than the one you show at eBuyer.
The FX-4170 scores 4776, gaming tier 3 CPU
http://www.cpubenchmark.net/...u_list.php
http://www.tomshardware.com/reviews/gaming-cpu-review-overclock,3106-5.html
The A8-6600K is about the same CPU performance at 4700 score.
The HD 6670 is considered a 2012 year entry graphics card, and scores about 1000
http://www.videocardbenchmark.net/gpu_list.php
The HD 8570D included in the CPU is not a separate card and scores only 681.
The Amazon system is better at gaming. It also shows a Corsair VS450 vs a 350 watt split rail FSP brand
http://www.realhardtechx.com/index_archivos/FSP.htm
for the ebuyer power supply.
Both have cheap computer cases, either the CiT Vantage or Cooler Master Elite 311

You can have a custom made from a good builder here:
http://www.dinopc.com/shop/pc/Raider-AMD-X6-6300-94p1640.htm
FX-6300, stock CPU cooler, your choice of WIndows 7 or 8.1, 8GB DDR3 1333mhz (2x4GB), 1TB HDD, R7 250 graphics, Zalman Z3 Plus case, VS450 PSU, and 3 year warranty for £475.60 without the games, possibly plus shipping, but this would perform 33% higher in gaming speed.
http://www.game-debate.com/...e-1gb-ddr3

You can choose to spend a bit more to move even higher in gaming.
http://www.ebuyer.com/570202-cyberpower-gaming-armour-lite-ecc01174
has hd 7770 and core i3.
It is 17% above the Dinopc
Amazon parts are
http://uk.pcpartpicker.com/p/2V6S2
£447 plus the games

The DinoPC parts would cost about the same, but uses better graphics.
custom @ pc specialist costs more.
http://www.pcspecialist.co.uk/computers/amd-trinity/