A countable set is one where you can reach any element in a finite amount of steps. See:

https://en.wikipedia.org/wiki/Countable_set

If you're saying we never get from the initial 0 to the trailing 1 in a finite number of steps, that's true and that's what DavidVoid is saying. But I haven't made the list of digits uncountably large by adding one element. Instead I put that element in a transfinite position in the ordering and I have to watch out for weird consequences.

I'm no expert, but countability doesn't depend on how the set is ordered. It depends on whether the elements can be placed in 1:1 correspondence with the integers. 1,0,0,... has a countable number of elements, and so does 0,0,0,...,1. They can be put in 1:1 correspondence with each other. This definition of countability is described in your link.

I meant to ask, does "countable representation" some kind of detailed definition that I can look at?

It's not the set of digits you use which is uncountable, it's the representation itself.

I think it's just another way to word DavidVoid's explanation a few comments up.