This is just a long comment.
There are a couple of things I don't fully understand in the question, and perhaps that's the reason why it is difficult to deal with (as the absence of answers testifies).

1. You write: "there is from the beginning an emphasis on a certain class of dynamical system being studied (e.g. smooth dynamical systems or group actions)". The problem for me here is the examples you give. Talking about group (and semigroup) actions is the most general/abstract point of view in dynamical systems. If even *that* is too particular for you, then it's difficult to do better.

2. Most of the literature on topological dynamics makes very simple assumptions once and for all: a (usually compact) metric space, a continuous functions, the action of $\mathbb{N}$ (or $\mathbb{Z}$). Any good book on the subject would be therefore a very well defined chapter within your desired general reference text. My two pennies: look at *Topological and symbolic dynamics* by Kurka.

What's wrong with composing yourself your "big text" gluing together a suitable number of books which are slightly more specific? Isn't it usual when dealing with such a gigantic subject? Would you search for an analogous book on, say, "geometry"?

3. Quoting again: "various viewpoints (dynamical vs. ergodic) and the definitions surrounding them aren't reconciled resp. contrasted". There are various excellent books in which, for instance, aspects concerning topological and measurable dynamics are carefully compared, but, again, they usually focus on one aspect (or a few aspects) rather than providing the most general theory possible. You can for instance look at [this][1] very nice book by Downarowicz, dealing with entropy (from both points of view). It may be useful in your perspective as the author tries in general to impose as little structure as possible on the phase space. For instance, the entropy structure is first presented considering the abstract theory of convergence of nets defined on abstract sets. 




  [1]: https://www.cambridge.org/core/books/entropy-in-dynamical-systems/668424EA998037F18673F9E002853047