I completely understand where you're coming from about feeling inauthentic. That's actually called "imposter syndrome", and talking down about yourself feels like a cure. In reality, self-debasement will make the imposter syndrome worse.
I suggest you adopt a different line of describing yourself. (This is the one I use myself):
Be as honest about your strengths as your weaknesses. For example: I am an expert in C++ memory management, but I have no skills in machine learning. I am fairly good at designing vector-based logos, but I cannot draw a decent stick figure to save my life. Keep your positive and negative self-statements in balance.
Own your mistakes, only allow yourself to apply negative terms to what you did, not who you are. For example: I once a wrote a scripting language in ActionScript 3.0 using regular expressions. It was terrible, slow, and I made sure to destroy that source code so it would never see the light of day again. I say: "the language design sucked." I don't say: "I suck at language design."
Avoid exaggerations. I'm a quantitatively verified expert in some areas of coding, but pretty dismal in others. Thus, I wouldn't say "I'm a programming expert" (unqualified), as that'd be an exaggeration; I'm only an expert in certain things. But I wouldn't say "I'm a mediocre programmer", because I've written some definitely non-mediocre things.
By the way, programming is art, so everything you told your students applies here. It's programming, not brain surgery. You're doing it! Never allow yourself to say "I'm not a real programmer." If you've written code, you're a real programmer.
Be honest about what you have yet to learn, but be just as honest about what you can do. Keep praise and criticism of self in balance.
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