Re: Can I color these pickup coils?
Don't look at it...or get used to it. Those are my real suggestions.
As for my suggestions that enable your condition:
If you want to paint them, I wouldn't use India ink. It's a mess, it has trouble sticking to and covering some materials, and it's relatively difficult to clean up. Use a paint instead. Something thicker, that can more easily be wiped or scraped from the areas where you don't want it. I would be tempted to try black acrylic paint (for art painters) or alcohol-based acrylic model paint, applied with a flat-sided, short-bristled hobby brush.
Here is a hint from model building: You might think that matte paint is the best black to use for painting holes, but gloss is generally better. It will appear as a darker, richer black, and it will obscure more of the detail, texture, and shape that you don't want seen. Matte black will show more detail, texture, and shape, and will read more as a very dark gray than as a pure black.
Acrylic paint should easily scrape off with a fingernail if you get some on the bobbins or the surrounding ring.
It used to be that Gunze-Sangyo was a great alcohol based model paint, but I don't know if it's still around, and it was quite the specialty item – hard to find in the days before the Internet, when I was using it. For your purposes, Tamiya model paint works fine, and it will probably be much easier to find in a local shop.
Tamiya paint:
https://www.amazon.com/Tamiya-Acrylic-Paint-Gloss-81501/dp/B0069FPPVE
The type of brush I'm talking about:
https://www.amazon.com/Tamiya-Mediu...d=1530121854&sr=1-7&keywords=hobby+brush+flat