:The most precise statement of the zeroth law is that thermal equilibrium constitutes an equivalence relation between thermodynamic systems. Unless you know beforehand what that means, its kind of opaque.  

 :Lets use  to mean "is in thermal equilibrium with". 

 :The Euclidean statement  is much more intuitively understandable, but is not, by itself, sufficient to establish the equivalence relationship. Only the property of reflexivity  is lacking. 

 :The transitive statement  is also not, by itself, sufficient to establish an equivalence relationship. Here, the two properties of reflexivity  and symmetry  are lacking. This makes it less precise than the already imprecise Euclidean statement. 

 :I don't know what you mean by "the usual statement". If the usual statement is "If two systems are in TE with a third, they are in TE with each other", then you have another Euclidean statement. It doesn't require symmetry to yield the zeroth law, it again requires only reflexivity. In fact, symmetry is rather implied in the statement "in TE with each other", but Euclidean and symmetry do not yield equivalence. While this statement might be attractive because it doesn't deal with all those A's and B's and C's, on the downside, it is not a clear statement, due to that vagueness about symmetry. 

 :This is the value of the Euclidean statement - it is clear and it requires a minimum of extra assumptions to establish the equivalence relationship that is the zeroth law. ONLY the equivalence statement allows you to assign a unique temperature to systems in mutual equilibrium. Any of the less precise statements deny you that ability. To say that the difference between a Euclidean or transitive statement of the zeroth law and an equivalence statement of the zeroth law is "nugatory" is to state that the zeroth law itself is "nugatory". You may say that the extra assumptions are intuitively and trivially obvious, but computers have no intuition, no sense of triviality, and if you are a computational physicist and ignore such distinctions, you will have garbage in and you will get garbage out.