The first "equivalent" means we can use the abbreviate function template in most respects as an ordinary function template. Such as explicitly specifying the specialization fun<T>(). The concept of the functionally equivalent is defined in [temp.over.link]. The comment is just a concise way to expose what "equivalent" means(i.e. expose-only). If we declare the abbreviate function template and the functionally equivalent function template in a program, it violates [temp.over.link] p7, I think.

The invented function template is used to depict what the synthesized declaration of the abbreviated function template looks like, such that we can explicitly specialize it or do anything identical to what can be done on a function template, without copying [temp] for it. If they were functionally equivalent, these points are not guaranteed. I agree we do not have a formal rule specifying what the comment says(i.e. functionally equivalent to the grammatical function template). We might just append a rule:

A function template declaration that corresponds to the function template declaration associated with an abbreviated function template is functionally equivalent to the abbreviated function template.

void g1(const auto*);

template<class T>
int g1(const T*);

such two function templates are neither equivalent nor functionally equivalent. As a contrast

void g1(const auto*);

template<class T>
void g1(const T*);

they are functionally equivalent, the program is ill-formed; no diagnostic required.

I agree we do not have a formal rule specifying what the comment says(i.e. functionally equivalent to the grammatical function template). We might just append a rule:

A function template declaration that corresponds to the function template declaration associated with an abbreviated function template is functionally equivalent to the abbreviated function template.

Is "what you can do with" a function template declared as an abbreviated function template really something that needs to be covered under "equivalence"? Will it not be suffice with "functional equivalence", whilst then reserving [temp.over.link]/7 and IFNDR for the case of conflicting declarations?

As a contrast

void g1(const auto*);

template<class T>
void g1(const T*);

they are functionally equivalent, the program is ill-formed; no diagnostic required.

With the current normative wording this program is not IFNDR (although I think it was intended to be), and we see compiler variance here where Clang seems to accept (although NDR...) abbreviated function template declarations as equivalent to (matching) function template declarations, whereas GCC does not (DEMO).

Will it not be suffice with "functional equivalence"

Not sufficient. We expect the "synthesized" declaration for the abbreviated function template is identical to what we said in the rule. Consider this example:

template<int N>
struct C{};
template<int N>
void fun(auto *, C<N>);

/*
template<int N, class T>
void fun(T*, C<N>); // #2
*/

we expect the synthesized declaration of the abbreviated function template like the one at #2, anything could be determined. If we say "functionally equivalent" to #2, that means you could have an infinitely possible declaration, such as

template<int N, class T>
void fun(T*, C<N + 1 - 1>); // #2

you can never explicitly specialize it like template<> void fun(int*, C<10>) because you never know which one the implementation would choose.

the program is ill-formed; no diagnostic required.

This means the document impose no requirement on the implementations, in other words, they can do whatever they want. See [temp.over.link] p8.

This issue seems a duplicate of #4121.

This issue seems a duplicate of #4121.

I think you mean "This issue is functionally equivalent to #4121".