Was there a time when one could have been just a software engineer building websites or did the specialization was there since the beginning of HTML and CGI servers?

From Google Trends [1], it looks like maybe we started using this term in 2010.

My personal recollection: Around 2006 we had new projects like Google Web Toolkit (GWT) and JQuery which enabled additional complexity in HTML/JS. In 2010, angular.js and backbone.js started and helped address a sentiment that JQuery was not good enough for large projects. It became hard for a software engineer to handle the new front-end frameworks while keeping up with the backend as well. So it would make sense to have the front-end engineer title introduced around this time. However, I don't recall if there were discussions or blog posts explicitly introducing it.

[1] https://trends.google.com/trends/explore?date=all&q=front%20end%20engineer,back%20end%20engineer