Congratulations! You have just discovered how poor these sites are. Many of the sites get this information from Alexa, though some try and discover the data for themselves. The fact that they do this is rather ridiculous. You cannot gauge a sites performance accurately or even close to accurately external to the site. Some exception exists for search engines which measure search performance or any site for which you have installed a bug such as Google Analytics. Even then, there are issues.
You are going to find a whole plethora of junk SEO sites trying to get your money with bogus data. Stay away from them. In fact, if you use these for SEO analysis, you will end-up all over the map and never really get any traction in the SERPs.
One exception I like is SEMRush. This seems to be the best out there and does not rely on scraped data from Alexa which is terrible or other sources. SEMRush does a good job at timely keyword SERP placements and values. As well, it is rather stable in it's analysis. While Alexa will fluctuate wildly, SEMRush remains stable and more trustworthy.
The downside to any of these is that no-one can gauge your traffic accurately and any keyword terms and rank/value for these terms are like looking into the rear-view mirror. They often just get the low-hanging fruit keywords from navigational links,
description meta-tag, header tags, etc. The real money keywords can be within the content itself.
The only worthwhile SEO online tools oddly come from Google. Use Google Analytics and Google Webmaster Tools at the very least. Spend time ignoring the hype and the bull and sit down and determine your own common sense solutions to better performance.
You edited your question after my answer so the ball has moved somewhat. Being fair, perhaps I should have commented on the sites you listed originally. The only reason why I added the answer is that my existing comment was only on-topic response at that point and I thought it should have survived. I will add some more specific detail though I will admit I do not follow these SEO tools around closely. The reason is simple. They are mostly if not utterly worthless. I have been at this too long to buy the bull they are selling.
Your examples fall into different product slots which is really quite a good thing. It allows this answer to cover the market fairly well.
Ahrefs does run it's own spider. It is actually fairly well designed and likely one of the better if not the best of the back link sites. I do occasionally visit this site, but it does suffer as many do from being grossly behind the truth. What I get from Ahrefs I can largely get from Google Webmaster Tools which offers a far more complete back link list, however, GMT is rather poor in how they present the data. Unfortunately, online sites do not offer too much additional value. Ahrefs I like, but it fits into that category. I would still advise people to visit Ahrefs from time to time.
I do not use this tool, however it is much like some others. I do use MarketSamurai. Generally, they are using a repository of data they lease from others. These generally come from websites that run their own bots similar to Ahrefs, OpenSite Explorer and MajesticSEO. These tools will change as different agreements are made. TrafficTravis, like MarketSamurai, can use your Google account to obtain keyword suggestions for you as well. These tools being downloaded applications, can make use of a variety of data sources. They mostly live and die on their ability to negotiate deals and make sense of freely available data. I cannot comment on TrafficTravis. It has never really broken the surface as far as high value SEO tools goes. However, people do like it. The problem with all of these tools is they are keyword centric which is not really helpful unless you like running around in circles. MarketSamurai and SEO PowerSuite (both of which I use) are the same but are at least the top two tools. The problem is that keywords is only a tiny portion of what SEO is about and these tools generally only really help with about 5% of the work. For keywords, nothing beats Google Trends and Google Adwords Keyword Tool. I found I get my best ideas from them.
ButtonSpace is a tool like ShareThis which I use. It is a bug that you place on your site. The data they have is based upon the activity your site has that they can track directly and any API available to them. I have not followed this world at all, however, social media sites are relatively open and share data well and make agreements easily. I get this data from Google Analytics though I do get e-mail notices from ShareThis. These services are good and I highly recommend them.
I do not make too much of a study of these sites mostly because they are pure junk. There are two methods used here. One is to use PHP to scrape a page directly and the other is to use an API to access data. Some sites get their data exclusively from Alexa and a few use AboutUs (which just uses Alexa for the most part) while others will also use other tools such as compete.com and blend it with some DNS/Whois data they gather themselves or get from another site. You can literally see how these sites piece together their data by watching various sites daily and watching the metrics change. You can really get a sense at how poor these sites are by doing that. These sites are built for one thing and one thing only- to make money either by selling you a junk service or by advertising. Nothing more. These sites are so inaccurate it is scary. But people still put their faith in them.
Trying to follow all this stuff becomes an addiction that you will excuse as being necessary. You will visit these sites daily even though SEO changes over months and years- not daily. You pour over them, try and make sense of them, analyze what you are doing right or wrong, fret and pull your hair out, gnash you teeth, and eat ramen noodles and drink diet dew all night long. The truth is this. It is a literal drug. It feeds the pleasure centers of the brain when in fact, none of this matters at all. It is like a heroin junky clambering for another fix when the drug does nothing for them. Slow and steady wins the race. Do not think a website knows your website better than you do. That is just plain silly talk!! Do not think a machine is smarter than you. Yes metrics are good and we need them. But for God's sake(!), please go to the source. You want to do well in search(?), then ask a search engine for help- not some shmo you do not even know. Well... except for me. ;-) I will pull your SEO junky butt (I said junky not chunky) out of the gutter and clean you up a bit.