Backlink confusion

Hi warriors.

There is something in link building that i cannot understand.

I use webmaster tools console, and some 3rd party seo tools like ahrefs and open site explorer to monitor my backlinks and (of course) to spy on my competitors.

The strange thing (which i believe everyone envolving to seo notices) is that all 3 programs show a different backlink profile. Ahrefs and Moz count links that do not appear in the webmaster tools console.and in general, every program shows it's own version of my link profile.

I understand that all 3 services use their own spiders to spider web, and it is kind of natural not to agree on the exact link list for my website.

I have also noticed that some links i have built on high authority websites like calameo.com do not appear in the webmaster tools console although my profile page on calameo is indexed in google's search results.

So the question is, when can you consider a backlink as a valid backlink you have earned?

Is there a chance that google counts backlinks that do not show in the console?
Is a backlink appearing on ahrefs valid when it does not appear in the console?
Is a backlink valid when the page containing it gets indexed altough the link is not counted in the webmaster console?

Thank you for your time.