I look at those definitions pretty much exactly in reverse. I cringe when somebody calls themselves a "coder" because my mental image is somebody who just throws it at the wall and stops as soon as it's good enough. (Many "enterprise types" certainly qualify.) "Programmer", to me at least, implies a greater level of methodology--not something that guarantees better code, but probably doesn't hurt.
I call myself a software developer, because I don't just write code. I build (develop) software. I hate the term "software engineer", because we're not engineers.
> I hate the term "software engineer", because we're not engineers.
I don't buy that. We're building software in a time before we have a universally accepted engineering doctrine for software, but you can hope that somebody calling themselves a "software engineer" is going to be using some kind of engineering-like doctrine.
Very, very few of us follow any sort of engineering-like practices. "Engineer" has real and meaningful connotations that almost no software developers follow or can follow.
Engineers have certifications and legal consequences to turning out a complete turd. If a PE signs off on something that catastrophically breaks, they can be legally culpable. With very rare exceptions, "software engineers" just file something in Bugzilla.
To me, developing software defines the entire process of delivering a software product. Looking at it this way, "writing code" is the only non-nebulous aspect a developer is tasked with. Domain knowledge and managing systems are very much part of a developer's role. Anyone abstracted from this is most certainly just a "programmer" or "coder." Real developers do a whole lot more.
Writing code is certainly part of it, but so is designing the overall software architecture, using the right library (when it already does a better job that your own "code"), keeping everything stable, and putting out a proper release. Each of these steps, and the dozens of others not mentioned, require one to "write code."
Programmers are often enterprise types, who don't code in their spare time, have a degree but don't learn anymore, and love the hours XML generates.
Many Indians-in-India code because their parents want them to be engineers, not because they love coding.