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"I am interested to know, apart from learning to program and developing/designing software solutions, what makes a software engineer an 'engineer'."

Nothing. "Software engineering" is something anyone can attempt to engage in. It seems to conjure up an image of something better than bad programming. But unlike, say, mechanical engineering, there is no professional certification required to call oneself a "software engineer". The fact that this hobby/job is an unregulated practice may have something to do with the ubiquity of bad software. Some of the best programmers, e.g., Arthur whitney, have referred to programming as being like "art". The programmer who started this site in fact wrote a book called "Hackers and Painters".