Question

Peano categoricity

Answer 1

You're not missing anything. One of the common criticisms of reverse math is whether the formalizations of notions really capture the informal notion.  One could certainly argue that their formalization doesn't fully do that for the reasons you say.

There's an increasing amount of work in "higher order reverse math" which attempts to address this concern.

Answer 2

Could you link the 2013 paper? Even if A is countable, it may not be a submodel of N, which is perhaps the more pertinent questions.