Without going into the details for the independent numbering, you can set differently the parameters for the spacing.
\documentclass{article}
\usepackage{amsmath}
\usepackage{lipsum}
\newenvironment{equation-}
{%
\abovedisplayshortskip=0pt plus 1pt
\abovedisplayskip=2pt plus 1pt minus 1pt
\belowdisplayshortskip=\abovedisplayskip
\belowdisplayskip=\abovedisplayskip
\begin{equation}%
}
{\end{equation}\ignorespacesafterend}
\begin{document}
\lipsum*[2]
\begin{equation-}
1\quad 2\quad 3
\end{equation-}
Short short short short short short short short short short short short
short short
\begin{equation-}
1\quad 2\quad 3
\end{equation-}
\lipsum*[2]
\begin{equation}
1\quad 2\quad 3
\end{equation}
\lipsum[2]
\end{document}

I'm not sure you really want it.
For the independent numbering:
\documentclass{article}
\usepackage{amsmath}
\usepackage{lipsum}
\newcounter{savedequation}
\newcounter{shortequation}
\newenvironment{equation-}
{%
\setcounter{savedequation}{\value{equation}}%
\setcounter{equation}{\value{shortequation}}%
\renewcommand\theequation{S\arabic{equation}}%
\abovedisplayshortskip=0pt plus 1pt
\abovedisplayskip=2pt plus 1pt minus 1pt
\belowdisplayshortskip=\abovedisplayskip
\belowdisplayskip=\abovedisplayskip
\begin{equation}%
}
{%
\end{equation}%
\setcounter{shortequation}{\value{equation}}%
\setcounter{equation}{\value{savedequation}}%
\ignorespacesafterend
}
\begin{document}
\lipsum*[2]
\begin{equation}
a=b
\end{equation}
\lipsum*[2]
\begin{equation-}
1\quad 2\quad 3
\end{equation-}
Short short short short short short short short short short short short
short short
\begin{equation-}
1\quad 2\quad 3
\end{equation-}
\lipsum*[2]
\begin{equation}
c=d
\end{equation}
\lipsum[2]
\end{document}

equationis standardlatexenvironment. – Zarko Jan 05 '18 at 23:18amsmathpackage — of which there are quite a few — in order to achieve your stated objective? (Theamsmathpackage modifies the behavior of theequationenvironment.) Why not just change the values of the length parameters\abovedisplayskipand\belowdisplayskip? – Mico Jan 05 '18 at 23:19amsmathredefined theequationenvironment. I want to keep the same compatibility. But if this detail is immaterial, ignore I mentionedamsmath. – Jan 05 '18 at 23:22equation*environment. – Jan 05 '18 at 23:38\vspace{-2pt}\[...\]\vspace{-2pt}if you want 2pt less than a normal display math – David Carlisle Jan 05 '18 at 23:41\abovedisplayshortskipis, by default,0pt plus 3pt– egreg Jan 05 '18 at 23:42equationenvironment either in plain LaTeX or withinamsmath. I think the problem of spacing may be due to your way of composing your equations on the one hand, and on the other hand to lack of use ofamsmaththat besidesequation, includes a lot of more environments to compose mathematics that surely you can adapt more easily to your needs. – Aradnix Jan 05 '18 at 23:52