What's the best way to express the idea of a dichotomy over 3 entities? Is trichotomy sufficiently clear, meaningful and correct?
Trichotomy is listed in Merriam-Webster. In mathematics one also uses this term when referring to the fact that a real number is either zero, positive or negative.
I think it's sufficiently clear and correct. Lots of dictionaries list trichotomy with the definition "division into three parts, elements, or classes".
Depending on the context tripartite ("in three parts") might be appropriate. It's certainly more commonly used than trichotomy in my experience.
You can still call dichotomy of three a dichotomy. It is a common mistake to assume that prefixes di/dia and bi are the same, where in fact they are not.
Di, in dichotomy means 'between or across' rather than 'two'. Dichotomy implies that your choice is limited between the options that you are presented with.
Another example where such an error occurs is dialogue, where people assume that if there is not exactly two individuals, then they can't have it. Some even try to make a false dichotomy by saying:
«It was not a dialogue, but a monologue»
Forgetting that a person can actually use monologue to communicate information between people.
A way to remember that di/dia does not mean two, is to think of a much clearer example of the term diagram. How many of these in here gram comes from graphein, meaning to make a mark or to draw. Thus diagram is something which shows interworking between several objects. And diagram of the circuit board can clearly have more than two gates (and we don't begin calling it trigram, quadragram, etc).