How to "center" a math operator defined by re-scaling a symbol?

Here's a solution that uses the \scalebox macro provided by the graphicx package. The solution employs the \vcenter "primitive" instruction to center its argument on the math axis.

I've also set the default scaling factor to 0.8, as 0.5 seems a bit extreme to me. (You are, of course, free to change the default scaling factor to suit your needs and preferences.) The scaling factor can be overridden "on the fly" for any instance of \smalltimes. This is useful if the symbol needs to appear -- not often, hopefully! -- in first- and second-level subscript or superscript positions. Since \scriptstyle and \scriptscriptstyle are 30% and 50% smaller than \textstyle, simply multiply your preferred scaling factor by 0.7 and 0.5, respectively, to get the proper sizing of \smalltimes for first- and second-level subscript or superscript material.

\documentclass{article}
\usepackage{graphicx} % for "\scalebox" macro

% Default scaling factor: 0.8
\newcommand{\smalltimes}[1][0.8]{%
    \mathbin{\vcenter{\hbox{\scalebox{#1}{$\times$}}}}}

\begin{document}
$a\smalltimes b + c\smalltimes d = e $

$\scriptstyle a\smalltimes[0.56] b + c\smalltimes[0.56] d = e$     % .7*.8=.56

$\scriptscriptstyle a\smalltimes[0.4] b + c\smalltimes[0.4] d = e$ % .5*.8=.4 
\end{document}

This answer is based on this answer.

\documentclass[12pt,letterpaper]{article}

\usepackage{lmodern}
\usepackage[T1]{fontenc}
\usepackage{scalerel}
\usepackage{amsmath}

\newcommand{\smallrel}[1]{\mathop{\scaleobj{0.5}{#1}}}
\makeatletter
\newcommand{\smalltimes}{\mathbin{\mathpalette\@smalltimes\relax}}
\newcommand{\@smalltimes}{%
  \raisebox{.4\height}{$\smallrel{\times}$}}
\makeatother

\begin{document}
    \begin{equation*}
        a\smalltimes b + c\smalltimes d = e
    \end{equation*}
\end{document}

Here is a version that is sensitive to the current style that you're in. Otherwise the rescaling behaves differently (for instance) depending on if you're in the normal sized maths, or in an exponent.

\usepackage{scalerel}
\newcommand{\smalltimes}{\ThisStyle{\scaleobj{0.63}{\raisebox{2.7pt}{$\boldsymbol{\SavedStyle\times}$}}}}

You may adjust the numbers to your liking!