This sounds like a contrived meta objection to me. Antisemitism-as-prejudice (footnote) lives in entirely in the brain of the perpetrator. The person you know was mistaken for jewish, in the mind of the perpetrator is jewish, and is being persecuted and killed for being jewish. EDIT: for that matter, a convinced Nazi would probably argue that no mistakes were made and if someone is “jewy enough to be mistaken for jewish, then they are jewish enough to be sent to the trains”. I’m going to immerse myself in betadine now for having written that phrase, yuck. Like in the perverted Nazi brain “jewishness” is a taint that must be aggressively expunged, regardless of niceties and nuances and stuff.

(footnote: as opposed to antisemitism-as-system)

I read and continue to interpret the definition as “discrimination of someone as a Jew for being of the quality Jewish” regardless of how Jewish they actually are.

The “Jews as Jews” to put the focus on the perceived identity in the eye of the discriminator.

To link back to your examples, people persecuted, murdered, by the Nazis and their allies for being perceived as Jews fits the definition as the motive for persecution was “Jews as Jews”, not for (pardon the stereotypes coming but examples of not Nazi enough viewpoints coming) “Jews as having curly hair” or “Jews as being Communists/traitors”, but the perceived Jewishness itself. Regardless of reality.