Your example code is not specializing, but overloading. The order does matter (not in your code though), because functions need to be declared before being known in C++. So if one overload calls another, or if another function in between calls the overload set, calls may end up somewhere not desired. Your example code is valid and common.
For some reason I think I remember reading that if you put the specialization below in this scenario then when the compiler looks through the header, it will see the default first, see that it works, and use that.
You are thinking of the following rule
If a template, a member template or the member of a class template is explicitly specialized then that specialization shall be declared before the first use of that specialization that would cause an implicit instantiation to take place, in every translation unit in which such a use occurs; no diagnostic is required.
I can't help but to quote the hilarious sayings of the Standard here about specializations
The placement of explicit specialization declarations for function templates, class templates, member functions of class templates, static data members of class templates, member classes of class templates, member class templates of class templates, member function templates of class templates, member functions of member templates of class templates, member functions of member templates of non-template classes, member function templates of member classes of class templates, etc., and the placement of partial specialization declarations of class templates, member class templates of non-template classes, member class templates of class templates, etc., can affect whether a program is well-formed according to the relative positioning of the explicit specialization declarations and their points of instantiation in the translation unit as specified above and below. When writing a specialization, be careful about its location; or to make it compile will be such a trial as to kindle its self-immolation.