Unsure why I have this version in google chrome (Version: 188.8.131.52) and the (184.108.40.2061) version in the Package manager. And why Firefox has the 220.127.116.111 verson?
pcpunk, I tried to explain this earlier, but since you didn't understand, I'll go about it differently, maybe this will help you to know.
What you see in the Package Manager has nothing to do with the inner components of Google Chrome, as it doesn't require Flash to be installed separate.
On the other hand, Firefox does, and that's why you see 11.2, 202.411 in there for it. Because natively, there is no way to get the latest Flash on any LInux install. The only way to have the latest Flash (and only on that browser), is if it's built into the browser, like it is with Google Chrome. Or any other browser that may include the latest. Firefox depends on the Flash installed on the OS, Google Chrome doesn't.
A few LInux distros has Google Chrome by default, for this reason. To give it's users the latest possible Flash content.
I can play the website video in firefox. At the Adobe site the newer version is (18.104.22.1681) http://get.adobe.com/flashplayer/?no_redirect Why would it not be the newer version?
As covered above, and as explained before, this is because you were checking on a Linux OS. Had you booted into your other OS & checked, you'd see a different result, that is, if Flash (or Adobe) are still supporting XP, not to worry, they won't be for much longer. Java announced their non-support status almost as soon as MS officially dropped support. Flash will be next on the hit list, and if Chrome decides to drop support for XP, game is over for those using that OS for any Internet content. It'll be the wide open gate for the least secure OS on the planet, AV installed or not, that the criminals are looking for to expose, as on Windows, Flash has to be kept updated, for security purposes. It's not the same with Linux.
However in either case, as long as Google Chrome will install, it won't matter if the OS supports it or not, it's there. The user cannot control this from the Package Manager, unless that person wants to uninstall Chrome, and why would they? Linux users are 4 major Flash releases behind, and numerous minor ones. For example, version 11 had ten or more minor "bug fix" releases (11.3, 11.4, 11.5, on through 11.9, you've missed out on these with Firefox), so did 12, 13, 14 & the latest version 15. If a Linux user didn't have Google Chrome, then versions 12 through 15 would have been entirely missed.
It's probably a minor issue that will be addressed in the next release, as a new one comes out like every couple of months. I haven't had this issue on either of my Linux Mint 17 with Google Chrome browsers. Maybe others has had this along with you, but I just booted and checked my other two computers, Flash content is working fine with Google Chrome on Mint 64 bit. If YouTube uses Flash, it's working on my end. Website videos may vary on any browser, I've had some to work, others not to. I don't know the differences, therefore nothing for me to explain, other than that I will disable add-ons (or extensions) to see if this helps. It's best to disable all when attempting this.
It seems that if I could install or replace the new plugin with the old plugin (22.214.171.1241) then it would work, but again this is just a wild guess and I can't figure out how to do this. Because Firefox uses the old plugin and it works.
In the meantime, why not just watch videos on Firefox & view the rest on Google Chrome? Unlike many other components of the OS which are open source, if those of Chrome are tinkered with in the about.config tab, you'll end up with a broken browser & need to reinstall again. You cannot install a 2+ year old version of Flash in a browser that upgrades almost monthly, just as Firefox does (unlike the default or shipped browser of the "other OS", that may get a new version once every 2 years, on supported OS's). If you've used Chrome for long (18 months or so), you'll have noticed by now that there's probably been more new versions of that browser (as well as Firefox) shipped, than in the almost last 20 years of IE, which was stuck at version 6 (IE6) for over five (5+) years. And how many releases of both Google Chrome & Firefox will come & go, becoming obsolete before IE12 is released?
That's what I do, is switch between browsers, using the one that meets the need. For example with me, it's downloading. The available add-on Down Them All not only downloads files faster than Google Chrome, but also has hash checking capabilities, provided one copies & pastes the correct value, and then select the type (MD5, SHA-1, etc.). Firefox delivers like no other browser with their native download add-on.
This won't be an issue for long anyway, as the bugs in HTML5 are being ironed out, and by 2016/17, Flash should be where it needs to be, in the history books (or Wikipedia), a relic of the "old" days of computing. If nothing else does in all of the older OS's, this will, as there has to be a content plugin for viewing. Surely Adobe will setup any computers that's connected to the Internet to auto uninstall what will be by then, an unlicensed to use app. Their EULA likely has the needed information.
As to why LInux users don't have the latest version of Flash all along, this is a huge question. Because it was Linux users & us only, that had 64 bit Flash two years before it was offered to Windows users. We used it, and one day with no warning (close or right before the version 11 releases), we no longer had the latest Flash content anymore. Adobe used Linux users as guinea pigs, then tossed us to the trash, once they had a usable Windows version. The exces there could have acted more with class over this matter, there was no thank you, [fill in the blank] you, no apology, nothing.
Adobe shafted the Linux community in a deep way & it'll never be forgotten by long-term users. It'll do little, if any, good for them to offer a "new, dynamic offering" to compete with the final version of HTML5 to us. As all of us long term Linux users remembers this, and it will never go away. Had Adobe acted with class, their actions may have been understood, like the folks who runs the Ubuntu family of distros did with Ubuntu One & other offerings, there was at least a 60 day notice in the matter. That wasn't a popular decision, but at least Canonical gave us notice.
Finally, that's why many of us has more than one browser installed, so that if one can't do the task, there's another that can. Simple as that.
The sooner Flash is gone, the better it'll be for all, one reason being security, the other being, it's a product that's behind the times. Mozilla, Opera & probably even Google all are working hard to deliver as much HTML5 content as possible to users & has been for some time. And like the way that Google Chrome is delivering Flash, HTML5 will be in the supported versions of browsers for all on supported OS's, not a backroom business deal cut with Microsoft to shaft those who doesn't run their OS (those who are running unsupported OS's are likely out of luck). Adobe will pay a hefty price for this mentality line over the course of years.
Even the late Steve Jobs was known for his strong distaste of Flash long back, and didn't ship it on Apple computers in later years. The end user had to install any Adobe software, if it was wanted.