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To my understanding it's more like Equal Temperament vs Just Intonation..?

No, those work even with sine waves. What the piano tuner is talking about is timbre: overtones can be not-perfectly-in-tune with the fundamental.

Has to do with the thickness of the core windings, I think? Something to do with the string acting partly as a rigid rod, not just an idealized wire. I hear this also in electric bass strings with super-heavy core wires.

Right, you're absolutely correct. I actually "get" inharmonicity, I think lack of sleep and the fact that tuning and the imperfect interval of an octave comes into it (I didn't know that tunings were actually shifted around inharmonicity) made me start thinking about JI/ET.. how embarrassing.

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