How should I react to seeing another student cheat on an exam?
You were right to inform the professor of cheating, and you shouldn't be penalized for that: your grade in the exam should depend only on what you wrote in your answers and nothing else.

However, I don't understand your motivation in writing the note. You should consider what you were trying to achieve by writing it. The way you describe it, the note sounds like it was lecturing the professor on how to do their job, rather than doing anything about cheating. You describe cheating as "unacceptable" but, by not naming the cheater, you show that you're more willing to accept cheating than the consequences of standing up to it. But, of course, it's your choice and your evaluation of the consequences is the most important.

You should perhaps consider other ways of dealing with the cheating. Other options include writing an anonymous note naming the cheater to the appropriate administrative staff. If you're worried that that would still leave you identifiable because the cheat knows you saw them, you could just start discussing the issue with other people in your class. If it becomes common knowledge that some unnamed person was cheating in the exam, peer pressure might stop that person cheating in the future.