I installed puma gem and when I start rails server by rails s I can see full output:
puma
rails s
$ rails s /Users/serj/.rvm/gems/ruby-2.2.1@email_
I've found this command in the Puma github readme:
$rails s Puma
Which also results in the desired behavior.
Edit: In fact, as described here after installing Puma, it should automatically be picked up by rails s command without the Puma argument.