I have been working in Mobile Acquistion for large mobile gaming companies for a while and can definitely understand the confusion, and sometimes frustration when trying to figure out the Mobile App ecosystem, especially as it relates to tracking and attribution.
To say there is a best one is subjective. MMP's or "mobile mediation partners" essentially act as a middle man to ensure that when you work with multiple advertisers that the appropriate advertiser gets credit for an install driven.
A lot of partners do this well. Adjust, Tune, Kochava, Fiksu, etc. However, I am bias because I work with MobileApptracking "Tune"as they call themselves now.
Mobile attribution is not an exact science and to say that FB Mobile App Install Ads are 100% accurate is incorrect. Typically you have to accept a level of a variance with nearly every sort of tracking solution. This ranges typically from 5-10%. Anyone who says tracking is 100% accurate doesn't have to look through the logs I have had to look through lol.
Facebook doesn't have a specific integration with the iOS/GP app store..you are actually integrating your app via the Facebook SDK, or via the Facebook API so it tracks in-app events including the install event. They are getting the install data from your app not the app store. It is fed via the integration you have setup with your app.
How Mobile Tracking Works:
If you understand the concept of ad tracking itself, you will understand that this is "confirmation." How mobile tracking works is like this:
User see's ad and clicks => redirect to MMP (on the redirect usually this is where device information is stored) => redirect to GP/iOS Store where the user may/may not download.
Please also keep in mind that you have integrated the MMP's SDK at this point!!
At this point a bunch of logs of clicks are stored for advertisers. Once a user has installed the tracking partner then is able to see where the install came from based on the click logs that were stored. Attribution then happens based on the default methodology. This methodology as is called last click attribution. So if Advertiser A,B,C all drove clicks but Advertiser C's click is the last. They would get credit.
The gentleman above me is also correct in that there are standard ID identifiers (IDFA, and GAID) that are collected in these logs to ensure that there isn't any breaking in attribution. There is actually an order of attribution in identifiers, but I don't want to confuse you here and Ill leave that for a diff. conversation.
If you are working with multiple ad partners there tends to be some overlap in inventory. So if you are running with 4 ad partners and all 4 have clicks from the same user, and one installs...all 4 will claim that install. MMP's help facilitate the attribution to the appropriate install claimer. This is called "de-duplication" Remember, again, you have their SDK integrated so they do have confirmation of an install event so there isn't "correlative" information it is confirmation of an install of your application.
IMHO opinion given the fact you may want to advertise across more than just Facebook, the best solution would be to integrate both the Facebook SDK, and a MMP's SDK. This will give you the flexibility to not only track Facebook accurately but to attribute across multiple certified MMP ad partners.
Given the fact that Facebook is very pricey at this point in terms of CPMs it makes sense to look for more affordable networks, exchanges, etc to acquire users.