What is the correct way to write a set of multiple variables or the multiples variables of a function in LateX 2e or with standard package?

Question

Which of it is the correct way in English typography, with LaTeX?

\documentclass{article}
\begin{document}
$f(x,y)= \ldots$

$f(x,\ y) = \ldots$

$f(x,\,y) = \ldots$

The coordinates $(x,y,z,t)$ \ldots

The coordinates $(x,\ y,\ z,\ t)$ \ldots

The coordinates $(x,\,y,\,z,\,t)$ \ldots

The coordinates $(x^0,x^1,x^2,x^3)$ \ldots $\leftarrow$ seems bad.

The coordinates $(x^0,\ x^1,\ x^2,\ x^3)$ \ldots

The coordinates $(x^0,\,x^1,\,x^2,\,x^3)$ \ldots
\end{document}

I write "seems bad" in the antepenultimate line because I feel same space before and after the commas.

Is there a package that automatize it? E.g a package which provide a command like \mycoord which we can use as \mycoord{x,y,z,t} and which add the correct spaces between comma in the output?

I'm aware of this question: Typesetting coordinates but answers are not clear about the spacing between the coordinate, and moreover, I don't want an answer with LaTeX 3, unless it's in a not so recent package (for compatibilities with a bit old system I don't control).

Edit: adding the comparison with a negative space before comma, in the case with variables with exponents.

\documentclass{article}
\begin{document}
1) The coordinates $(x,y,z,t)$ \ldots

2) The coordinates $(x^0,x^1,x^2,x^3)$ \ldots 

3) The coordinates $(x^0\!,x^1\!,x^2\!,x^3)$ \ldots
\end{document}

Does the third seems better than the second?

Second edit:

How I see the spacing (probably wrongly..., but the third seems a little better than the second, but perhaps the negative space is a little too big) :

Answer 1

The package semantex (disclaimer: I am the author) can do many things like this in a customizable way. Unfortunately, it is a relatively recent package, but I believe it should work with older versions of LaTeX3, too, though I have not checked this.

\documentclass{article}

\usepackage{amsmath,semantex}

\NewVariableClass\MyVar

\NewObject\MyVar\mycoord[
    left par=(,
    right par=),
    set arg sep={,},
    set arg dots=\dotsc,
]

\begin{document}

$ \mycoord{x^0, x^1, ... , x^p} $

\SetupObject\mycoord{
    % Let’s change the settings:
    set arg sep={,\,},
}

$ \mycoord{x^0, x^1, ... , x^p} $

\end{document}

---

Otherwise, for a more portable solution, you can try editing this answer by egreg. As he notes in the comments, the comma can easily be replaced by anything else. And I believe this should work perfectly well with much older versions of LaTeX3.