If you are using the classes from the second example in more than one place in your HTML and then depending on your actual CSS in those classes, then the second option may be more efficient then your first. In SASS, it will totally depend on how you write the actual CSS, not just the class versus id debate.
But a question from the article comes to mind in your example, what if you are requested to insert a second login widget to the page? Even worse, if your Javascript uses the ID for targeting the login code then you have to reconsider that code as well.