In my experience "note" is indeed used for both written and perceived pitches, though "tone" in the latter sense might appear in a textbook on harmony or acoustics. In a pop or jazz review, I would expect "tone" more for general sound quality or character: "His horn had a warm, sweet tone."
A few more options for "Nebenton" might help. As soon as overblowing was mentioned (though IANA wind player), my first thought was "harmonic" or "overtone." Could "Nebenton" be just a variant (more colorful, less technical) of "Oberton"?
One meaning of "Nebenton" is "secondary accent," as in word stress. In English, "accent" can be used figuratively: "a splash of bright blue provided a cheerful accent." Colorful language is often used in describing music. Given the fireworks metaphor, would something like "(counter-)accent," "splash," or "flourish" fit?
Or in the context of a run or scale, tones/notes outside the main chord structure can be "neighboring," "dissonant," "nonharmonic," etc., and often do provide a kind of counter-accent to the primary harmony, much as intentional violations of the meter might in a poem. In nontechnical language, "neighboring notes" might do, or maybe even "extra notes."