Defining decision-problem complexity classes by counting branches of a polynomial-time NTM
In the abstract of Counting classes: thresholds, parity,
mods, and fewness”, by Beigel and Gill, any class of languages whose membership of some input $x$ can be determined by some function of $\#M(x)$ (the number of accepting paths of the NTM $M$ on input $x$) is considered a counting class.

In regards to your "counting description" of some known classes, your characterization of $\mathsf{ZPP}$ is actually a description of $\mathsf{P}$ (there is no mentioning of the power of expected polynomial time over worst case polynomial time). Additionally, your description of $\mathsf{P}$ is not very clear, a Turing machine can be either deterministic or non deterministic (those are different models of computation), so there is no NTM which is "actually deterministic". It is not clear how to give a counting characterization of $\mathsf{ZPP}$, since it differs from the rest of the classes by relaxing the polynomial time property.

Additionally, note that in the NP/coNP case you have an equivalence, i.e. $x\in L$ iff there exists some accepting path/ iff all paths are accepting. Requiring only one implication results in a stronger class.