IMHO programmer, coder and software developer are different words for the same thing. Do you write code, program a computer or develop software? Or are they actually the same thing?
A programmer create algorithms and structures but without using a specific language. Maybe using things like pseudo code or "Jackson Structured Programming" or etc. A coder will take the work of a programmer and turn it into usable, compilable, code written in a language. They are often the same person. Often the "programming" is done at the same time as the "coding".
You are the only person Ive talked to who has such a strict definition for the terms. Everyone else seems to either use them interchangeably or has very loose definitions.