There are many reasons for Google not showing your back links and one of them is indexation. As you build back links to your site, to get it ranked and properly indexed. In same way back links also requires further back links to get them indexed.

Here comes the power of Tier link building. Go for wide and extensive tiers for proper indexation of your back links and squeeze link juice properly.

While Registration webmaster tool will provide you link you just copy and inserted in your on page site after that there will be several options like crawl error, sitemaps and many more check that all if any error is showing then you need to correct the errors..

If you are not getting your backlinks in google webmaster tools, just check how much pages of your website has been indexed by google and also try sharing your links on social networks and different profiles you own. This way your links will get indexed and there would be greater possibility for your links to show in google webmaster tools.
Meanwhile, you can also see some free tools available online for backlink checking.

ahrefs,webmaster tool,majestic all shows different amount of backlinks.. I don't know which tool is correct.