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If you are speaking to another programmer you may be able to speak about you technical skills and you achievements.
I mean the projects you had challenges when working on.
You may be also able to speak about your portfolio and your vision on your carreer.
That I'm not sure. May be, in my case, I would start with my name then describe my programming background: programming languages, my proficiency or profession with them, and how long I have been using for which field.
The best way to introduces your self as programmer to another programmer to speak about your real skill and give a creative example of a job you can do, and the simple method to do it(Demo), because any programmer know Coding main rule is that "make your code simple as you can, with less line of code".
if you ask me how a programmer should present himself to another programmer:
I would say that first of all, you must present yourself correctly (name, age, sex...).
then start talking about your programming history, coding, the project you have done and the languages you speak.
You may start with your name and tell him/her your breakthrough codes you have written, your current project and seek advise/help on your present challenges.
First, present himself (name, age, nationality, address, ...) then he can start describing his coding experience and skills he had acquired.
He can use hosted projects on Github for ex, or make a demo using live projects (web, embedded systems, mobile applications, ...)
The best way to introduces your self as programmer to another programmer to speak about your real skilland start with your name and tell him/her your breakthrough codes you have written.
He speaks of himself in the first and provides the languages proficient in the field of programming and then offers the various projects done and then his vision through which clear objectives
the programmers is a humans also , everything will be normal .
maybe we will talk about our skills and programming technology