As the question implies, I have two main research interests in the natural language processing and computational genomics domains. While I have a preference for the former, I am also very interested in the latter. I ask because I am not sure if it would look like I am too unfocused for a PhD. Many professors seem to have multiple areas of interests, some of which are not too related, but nonetheless I want to make sure that mentioning multiple interests will not hurt me.
If you can convey your enthusiasm for the overlap of these two somewhat distinct areas, you will be ahead of the game. A PhD candidate is expected to extend the limit of human knowledge, and one of the better ways to do so is to research within the overlap of two (or more) disciplines. I would tend to see having more than one area of interest as being an advantage rather than a disadvantage. Of course, as your question implies, a multiplicity of interests can backfire by causing a candidate to appear unfocused. Focus on the overlap, and make your application shine!
Your PhD should be somewhat focused, but before that, you're not in general supposed to have real scientific results. I would surely mention both, unless you feel that one of them is significantly unsignificant.
If you managed to have multiple results as a master student, that makes you a very good candidate for a PhD, because not many people have that!
The answer depends on you, the professor and when and how you communicate, .
Better mention both than fear. Until you handle the fear to stand for both interests it will not go away. When you are clear about what you fear you may know the answer already.
Showing that you have a broad interest means that you are open to change.
You may get angry at yourself for not having mentioned the other interest. Especially because professors know other professors to redirect to.
If you write you can make sure that it is clearly understood that you have two interest. Such as by choosing a headline for each. The professor may then choose what to focus on.
If you talk to the professor you get responses what (s)he likes most. Then you can focus on that.
When there is not enough time for both topics it could be bad to mention both.
Mentioning both can mean that the professor does not choose you because (s)he only takes in people with one interest who can be put under more presure because they can not choose to go somewhere else.
Sometimes professors are really proud of themselves because nobody ever tells them they are wrong. You may have to adjust what you say so they do not fear your intelligence in a field they do not know.
I come from a computer science background. There may be different cultural influences in other professions.