I figured what was happening. Turned out that I did not assemble my new camshaft adjuster properly when I was installing it. In a very simplified way the adjuster contains of the inner rod, which is attached to the camshaft, and the outer gear. The outer gear is supposed to be put on the rod all the way in, and this was something I failed to notice when I was disassembling the old adjuster. The outer gear was holding to the inner rod by some tiny margin of the tooth. Because it wasn't put on the rod all the way in, the remaining gap had prevented the magnet to operate properly, and that was triggering the engine light. I've attached two photos, before the fix, and after. Notice the gap between the two red arrows when I installed a new adjuster for the first time.
Luckily I practically didn't drive the car until I figured it out. I still don't understand how the engine is survived. If the outer gear had slid or break that tiny piece of the inner rod's gear teeth, it would freely rotate without turning the intake camshaft. Not something you'd like to happen.
But that's not all. I've reassembled everything back in the proper order, and, to my big suprise, the engine started rattling again after a few days. It didn't rattle as much as before, but still, once or twice a week I was hearing that annoying sound. And that was happening until I've decided to change my engine oil. After the oil change, the rattle disappeared. I did not hear it even for a single time.
I always put the same oil (Mobil 0W40). When this rattle has started, quite some time ago, I've checked the oil level among other things. I even topped it up a bit to be sure.
So what do you think? Can the oil be a problem? Bad batch? Contaminated? I did not change any other fluids for some time. I put Castrol oil in my wife's car, so I don't think that I just poured another can by mistake.

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2002 C230K