In looking around on the web, I realized that UI Engineer is a poorly defined job. Additionally it's good to be able to explain one's own position. So, here is my take.

A user interface engineer (for the web) is a person that codes functional, usable, accessible, and hopefully enjoyable software that runs in your web browser. These folks work closely with, or are themselves, conceptors and designers of said software. They also have a keen sense of the technological landscape, and constraints, that inform the solution written to support user actions through a web site.

Back to an opening statement about me.