£600-£650 gaming PC build

It's not great, there's quite a lot you can improve on for the money.
The issue is that if you aren't building it yourself then you'll need to go with whatever components your chosen builder has available (unless they'll let you pick stuff out) which makes it much harder to suggest improvements.

As a general suggestion -
Get the cheapest i5 you can
Avoid Z-series motherboards and focus on getting a quality H (or even B) series which fulfils your requirements.
Stick with the stock CPU cooler
Go with the minimum HDD you think is sufficient for your needs, this is a really easy thing to upgrade down the line.
Find a quality PSU (the one listed is not very good, if you have a list of options toss them in here).
Go as cheap as you can on 8Gb of ram.
Push as much money as you can into graphics.

Also - are you sure this is £650 at CyberpowerPC? I had a glance at their configurator and it looks like you are looking at the total without VAT added. (Why they do this for a consumer PC market I really don't know).

Yeah the 270X is probably plenty sufficient, my comments were a very general comment on how you might break down your budget.

In general yeah, go for 2*4Gb, the price difference is usually negligible. Whoever told you the boost is 10% is speaking nonsense though, it varies dramatically depending on the application and for gaming it's not much at all.

If you have a friend who can put things together for you, or you fancy a challenge yourself, then you can make your money go further and perhaps more importantly, avoid some of the "fails" of companies like that.
Here's a fairly random example of what you should be able to afford if you pick components yourself. Obviously the availability of stuff will change if you go into a local shop and ask them to knock up something similar, and it'll cost you more, but I'd imagine a similarly specced machine could be done by a shop for around £650.

Well if you have any doubts about stuff just ask away, either here or make a new thread in a suitable sub-forum.

PSU is a tricky one. Most retailers (like Cyberpower) will just stock a couple of options at each wattage level and they won't necessarily be there because they are good items. If you are going with a stock-clocked i5 and an R9 270X, then a 500-550W PSU is likely your best bet (ignore Cyberpower saying you need a 600W).
The Corsair CX you picked in your 2nd attempt is a fairly mediocre PSU, it's likely fine for you but it's not the best quality.
The XFX I picked out is a very solid, budget minded PSU so if it's an option then it's a safe bet (as is basically anything from XFX/Seasonic).

Cooling is actually very straightforward. If you have a stock speed i5, then the standard Intel cooler will work fine enough unless you live somewhere incredibly hot (in the UK you should be fine lol).
If you can pick a decent entry level case, ideally one which comes with a couple of fans as standard, then it's likely you'll be fine without any upgrades. On a basic system like this just having an intake and an exhaust makes all the difference.
It's also pretty easy to tweak later, even if you aren't too confident with fiddling with components.

Motherboards get very complicated as each chipset (Z97/H97 etc) has it's own set of features, then manufacturers chose which of those features to include, which to remove, and which additional features of their own to add. This is how they build up their product lines, but it's a pretty baffling array of letters, numbers and acronyms to most people.
The quick (and over-simplified) version -
Z97 - Overclocking+SLI, added features like SATA Express and M.2 support.
Z87 - Overclocking+SLI, effectively replaced by Z97 now
H97 - Updated features like SATA Express and M.2 support.
H87 - Mainstream board, effectively replaced by H97 now.
H81 - Entry level board, only 2 memory slots, usually limited SATA connectors too (as well as other features).
B85 - Officially speaking a "Business" board but falls somewhere between H81 and H87 in features (less likely you'll come across these)

Basically if you aren't overclocking, a H97 is a good option looking forward as it has connectors for new technology which may or may not catch on. Some boards won't have all of these (see above) but if you want a solid option which isn't lacking in features and is likely to have enough connectors for you to (get someone to) upgrade later then they are generally a good bet regardless.

Nobody needs a modular PSU really. In some cases they can be useful if you need to remove the extra cabling, but it's mainly unnecessary. If your case has some cable management (like cut-outs in the motherboard tray) then it's incredibly easy to get all the cables tucked away nice and neatly.