Why are [some] professors ambiguous about material that is actually tested on exams?
There are at least four worries, in my experience:
The students may be trying not to study any more than they absolutely have to study. So if the professor says something isn't on the exam, they won't study it.
If the professor says what is on the exam, but doesn't explicitly mention something, and then that thing comes up (even as a minor part of another problem), the students may complain that "you said that wasn't on the test!". This can happen even if the professor really made a good faith effort to say what was on the test, and the students simply misunderstood.
The professor may not have written the exam yet, and so she doesn't know the exact topics that will be included.
Someone else may write the exam, if there are many sections of the course taking the same exam. In that case, the professor may not be permitted to say what is on the exam. When I was a postdoc, they didn't even show us the common calculus exam until just before it was administered.
There are various strategies to cope with these worries. A common one, as described in the question, is to just say that the exam can include everything from the class, which is not very informative but is otherwise harmless.
There are other strategies, as well, such as making an exam review packet that includes more than the exam possibly could, and then selecting exam problems based on the review packet. But these don't help with the issue of common exams written by someone else.
By the way, if turnaround is fair play: we professors often ask the dual question: why do students so often ask what will be on the exam, when they have just had a class on the same material that will be on the exam? As you can imagine, we may feel that we have already told the students what we want them to know, by designing the course to include it!
Due to time constraints, most exams directly ask about only a small fraction of the course material. If a professor explicitly tells students exactly what parts of the material are going to be directly tested on the exam, many students would only bother to learn that material.
Therefore, professors often include anything they want students to learn/study in the exam coverage.
(There are obviously tradeoffs involved: include too much and students won't be able to study the really important things at a sufficient level of depth, include too little and the students won't get enough breadth.)
There are several aspects to this:
First of all: I don't know what questions I will use in the exam before I actually have written it - which may very well not have happened before the actual day of the exam. I have a pool of over 300 questions, categorised by the expected length of the answer (no multiple choice over here in Switzerland - you must word your own answers). The duration of the exam (normally 4x60 mins) determines the number of questions from each category. But the actual questions are chosen randomly, I even wrote a programme to do this part for me, to make sure that it is fair (I have my "favourites", the programme doesn't).
Secondly, I want the students to have a close look at large parts of the most important materials of the field. I don't expect them to know everything and all questions are worded in a way, that you can answer with the knowledge from one area or that of another; e.g. a question on the relevance of Hegel or Nietzsche can be answered from a historical point of view (what lead there and where did it lead afterwards?) or from a philosophical point of view (what did they say and why is that important?). So if you're weak in historical knowledge but strong in philosophical, you can put the stress on the latter and still get full scores - and vice versa, of course.
This way the exams are fair: Everybody knows the area, they can learn what is most interesting to them and still everybody can get good grades, given they really put effort into the preparations.
For oral exams the answer is pretty similar: The examiner and the co-examiner prepare a pool of questions at a meeting normally not much more before the exam than maybe a week. So also here: We simply don't know what questions will be in the exam, we just know which areas we want to cover. Also here we will adjust the questions to the strongest areas of the examinee. They have 5 to 10 minutes at the beginning to show us what they know: They choose a topic and get started. After that we ask questions with stress on that area. You quickly know which students have prepared well and which didn't.
Basically: If we see the effort, the grade will be good.