I'm curious, am I completely off the mark? Do you see yourselfs as artistic when you code?

Yes, absolutely. That's precisely the reason I consider myself a craftsman (if I may say so myself). As someone else said in an earlier post, crafts are something of a mixture between science and art. I think this is true, and what you say reinforces it in my mind.

The point you bring up is interesting, though. I suppose the choice between being primarily a "craftsman" and an "engineer" is entirely personal. I guess it's obvious and true, but just something I hadn't really put any thought towards.

I truly believe that the 'outside' world does not understand programming, or why we get such a kick out of it.

I agree. This is something I had in mind while writing my original post, but I'm not sure it came across quite like I wanted. This is, I think, a common factor of most crafts -- people on the outside see a programmer, or a carpenter, or a mason, but do not necessarily think of all the differences between different programmers, carpenters, and masons. This is the thought that caused me to post in the first place.