Earlier quoted context omitted.
The same argument used to call a programmer an engineer could be applied to anyone that writes for a living, any doctor, any lawyer, well pretty much anyone. We're all "engineers" unless we're doing something completely mechanical.
According to the dictionary, an "engineer" is a person who designs, builds, or maintains engines, machines, or public works. A computer is a physical machine that runs programs which are encoded digitally on physical hardware (memory). I could see an argument for considering a computer program to be an engineered "machine" that runs on a turing-complete computational platform. Just because a computer program is digit…
That's exactly the point. Programmers write programs. They don't design engines, machines, or public works. Programmers are excluded from the very definition you're quoting.
If you want to expand the definition to include the application of scientific methods to solve a problem, so that it includes someone that creates a Javascript single page web app, there's no way to exclude those other professions. Programs are not machines by any conventional use of the term.