When I type one of my site name, like "example" (that is actually example.com) I see title as "example" in my google search result, while I should be seeing the actual site meta title. It shows "example" and then followed by few popular links under that title.
While some of my sites returns actual meta site title but few only "site name"
I assume you mean the <title> element - and not a meta tag named title. If that is true, then it is also true that Google has begun to revise titles and descriptions snippets as well, depending on the search terms entered.