Is there a clear definition of "computable" for models of computation which are not turing complete?
Some basic fact that you are missing here is that all the encodings that you mention are equivalent from the perspective of computability: there is a computable function mapping the binary encoding of a number to its unary encoding, or vice versa. Therefore for the sake of defining computability, it does not matter which of these encodings you choose for numbers. Just fix your favorite encoding.

Computability is at its core a property of string functions $f\colon \Sigma^* \to \Sigma^*$. When you define computability in any other domain, you have to fix an encoding. In practice, all "reasonable" encodings are equivalent in the sense of the preceding paragraph, so the exact encoding doesn't matter.

The encoding does, however, matter in restricted models of computation. To take an extreme example, suppose that you consider time-restricted Turing machines: say you want your machine to terminate in time $O(n^c)$ for some $c$, where $n$ is the length of the input (as a string). We can no longer switch between binary encoding and unary encoding, because binary encoding is much more compact. When we talk about a polynomial time computable function of integers, we specify that integers are encoded in binary. Even this is a somewhat arbitrary choice, since decimal encoding would lead to the same notion of polynomial time computability.

So to answer your question – the encoding is specified as part of the definition of the restricted model.