I'm not entirely sure if this is on topic or not. I mean, on the one hand it is because it has to do with testing. On the other, well, I'm not sure if it's actually answerable - but I'll give it a shot.
Whether you call yourself a developer or programmer or coder or software engineer is entirely up to you. You can put anything you want on your resume and it's up to the person who interviews you to do their due diligence to determine if what you said you did matches what you did. As far as I'm aware, no country has a board you have to apply to to put titles on your resume (although, I know using the word "Engineer" can have some rules attached, I have never heard of "Developer" or "Programmer" having the same level of scrutiny.)
If you feel what you do is an Automation Engineer, feel free to put that as your title on forms, work profiles, etc. If you think your boss would have a problem with it on official documents internally, well you will have to take that up with them.
It is certainly true that many automation engineers do a ton of programming in their efforts, and the only people I think saying you couldn't ever say you were a "programmer" would be junior programmers who are jealous that you'll steal their job - pay them no mind!