I would say I consider myself an Engineer.
How you would apply term "Programmer" to a Front-End development with all CSS and HTML markup ?
About word "Software Engineer" - well I know Car Engine, Motor Boat Engine, Aircraft Engine. And sure we also have Game Engine, Web Search Engine......
Well if we use Engine also for that code stuff, then yes "Software Engineer" is a fair term for specific niche code/virtual product development/building/engineering.