How add new argument to command to control scaling?

Question

The following code, slightly modified from https://tex.stackexchange.com/a/552769/13492, allows creating new symbols \bigplus, \bigtimes, and \bigbox.

However, it hard-codes the scaling factor \scalefactor used to determine the horizontal shinkage of the symbols as produced, in the first instance, by applying in order \my@makebig and then \my@big to the pictures created by applying \my@draw to the basic forms \my@plus, \my@times, and \my@box.

The hard-coded \scalefactor is controlled by \ifnarrow.

Question:

How can the code be modified so as to use a new, additional, argument to specify the scaling factor directly and separately for each individual symbol (and such other similar symbols as I may choose to create).

Ideally, that new argument would be optional, defaulting to 1.

I just don't see which of the commands to modify how so as to accomplish this.

Clarification:

The commands \myplus, \mytimes, and \mybox are to be left with each having no arguments. Rather, a new argument — should be added to \my@big, \mymakebig, and/or \my@draw (whichever are needed for the purpose). The new, optional, argument is then used once, in the preamble, separately for each of the \big... commands. In other words, to allow something like this in the preamble:

\DeclareRobustCommand{\bigtimes}{\DOTSB\my@big[0.8]\my@times}    
\DeclareRobustCommand{\bigplus}{\DOTSB\my@big\my@plus}

And then in the body of the document I still use, e.g.:

$\bigplus_{i\in I} X_i  = \bigtimes_{i\in J} Y_i$ 

The code now:

\documentclass[fleqn]{article}
\usepackage{amsmath,pict2e,graphicx}
\usepackage{iftex}
\ifTUTeX
  \usepackage{unicode-math}
  \setmainfont{TeX Gyre Termes}[Scale=1.0]
  \setmathfont{TeX Gyre Termes Math}
\else
  \usepackage{newtxtext,newtxmath}
\fi
\newif\ifnarrow
\narrowfalse
\makeatletter
\newcommand{\my@big}[1]{%
  \mathop{\vphantom{\sum}\mathpalette\my@makebig{#1}}\slimits@%
}
\AtBeginDocument{%
  \DeclareRobustCommand{\bigplus}{\narrowfalse\DOTSB\my@big\my@plus}%
  \DeclareRobustCommand{\bigtimes}{\narrowtrue\DOTSB\my@big\my@times}%
  \DeclareRobustCommand{\bigbox}{\narrowfalse\DOTSB\my@big\my@box}%
}
\newcommand{\my@makebig}[2]{%
  \ifnarrow
    \def\scalefactor{0.8}%
  \else
    \def\scalefactor{1}%
  \fi%
  \vcenter{%
    \sbox\z@{$\m@th#1\sum$}%
    \setlength{\unitlength}{0.9\dimexpr\ht\z@+\dp\z@}%
    \hbox{\kern0.1\wd\z@\scalebox{\scalefactor}[1]{\my@draw{#1}{#2}}\kern0.1\wd\z@}%
  }%
}
\newcommand{\my@draw}[2]{%
  \begin{picture}(1,1)
    \linethickness{%
      \ifx#1\displaystyle 1.15\fontdimen8\textfont3\else
      \ifx#1\textstyle 1.05\fontdimen8\textfont3\else
      \ifx#1\scriptstyle1\fontdimen8\scriptfont3\else
      1\fontdimen8\scriptscriptfont3\fi\fi\fi
    }%
    #2
  \end{picture}%
}
\newcommand{\my@plus}{%
  \roundcap
  \Line(0.5,0)(0.5,1)
  \Line(0,0.5)(1,0.5)
}
\newcommand{\my@times}{%
  \roundcap
  \Line(0,0)(1,1)
  \Line(0,1)(1,0)
}
\newcommand{\my@box}{%
  \roundcap
  \Line(0,0)(1,0)
  \Line(1,0)(1,1)
  \Line(1,1)(0,1)
  \Line(0,1)(0,0)
}
\makeatother
\begin{document}
In-line $
\bigplus_{i\in I} X_i  = \bigtimes_{i\in J} Y_i = \bigbox_{i\in I} X_{i} = \bigcap_{i\in I} Z_{i} 
=  \bigcup_{n\in I} Z_{i}  = \sum_{i \in I} H_{i}
$. Display:
[
\bigplus_{i\in I} X_i = \bigtimes_{i\in J} Y_i = \bigbox_{i\in I} X_{i}= \bigcap_{i\in I} Z_i 
=  \bigcup_{n\in I} Z_{i} = \sum_{i \in I} H_{i}
]
In-line $
\bigplus_{i=0}^{\infty} X_i  = \bigtimes_{i=0}^{\infty} Y_i = \bigbox_{i=0}^{\infty} X_{i} 
= \bigcap_{i=0}^{\infty} Z_{i} =  \bigcup_{n=0}^{\infty} Z_{i} = \sum_{i =0}^{\infty} H_{i}
$. Display:
[
\bigplus_{i=0}^{\infty} X_i = \bigtimes_{i=0}^{\infty} Y_i = \bigbox_{i=0}^{\infty} X_{i} 
= \bigcap_{i=0}^{\infty} Z_i =  \bigcup_{n=0}^{\infty} Z_{i} = \sum_{i=0}^{\infty} H_{i}
]
\end{document}

Output: This is what xelatex produces; the output from pdflatex is similar.

Related: The issue arises from my original question Make big plus and big times symbols and the answer https://tex.stackexchange.com/a/552557/13492 to it.

Answer 1

REVISED, based on comment.

\documentclass[fleqn]{article}
\usepackage{amsmath,pict2e,graphicx}
\usepackage{iftex}
\ifTUTeX
  \usepackage{unicode-math}
  \setmainfont{TeX Gyre Termes}[Scale=1.0]
  \setmathfont{TeX Gyre Termes Math}
\else
  \usepackage{newtxtext,newtxmath}
\fi
\makeatletter
\newcommand{\my@big}[2][1]{%
  \mathop{\vphantom{\sum}\def\scalefactor{#1}\mathpalette\my@makebig{#2}}\slimits@%
}
\AtBeginDocument{%
  \DeclareRobustCommand{\bigplus}{\DOTSB\my@big[.5]\my@plus}%
  \DeclareRobustCommand{\bigtimes}{\DOTSB\my@big[.75]\my@times}%
  \DeclareRobustCommand{\bigbox}{\DOTSB\my@big\my@box}%
}
\newcommand{\my@makebig}[2]{%
  \vcenter{%
    \sbox\z@{$\m@th#1\sum$}%
    \setlength{\unitlength}{0.9\dimexpr\ht\z@+\dp\z@}%
    \hbox{\kern0.1\wd\z@\scalebox{\scalefactor}[1]{\my@draw{#1}{#2}}\kern0.1\wd\z@}%
  }%
}
\newcommand{\my@draw}[2]{%
  \begin{picture}(1,1)
    \linethickness{%
      \ifx#1\displaystyle 1.15\fontdimen8\textfont3\else
      \ifx#1\textstyle 1.05\fontdimen8\textfont3\else
      \ifx#1\scriptstyle1\fontdimen8\scriptfont3\else
      1\fontdimen8\scriptscriptfont3\fi\fi\fi
    }%
    #2
  \end{picture}%
}
\newcommand{\my@plus}{%
  \roundcap
  \Line(0.5,0)(0.5,1)
  \Line(0,0.5)(1,0.5)
}
\newcommand{\my@times}{%
  \roundcap
  \Line(0,0)(1,1)
  \Line(0,1)(1,0)
}
\newcommand{\my@box}{%
  \roundcap
  \Line(0,0)(1,0)
  \Line(1,0)(1,1)
  \Line(1,1)(0,1)
  \Line(0,1)(0,0)
}
\makeatother
\begin{document}
In-line $
\bigplus_{i\in I} X_i  = \bigtimes_{i\in J} Y_i = \bigbox_{i\in I} X_{i} = \bigcap_{i\in I} Z_{i} 
=  \bigcup_{n\in I} Z_{i}  = \sum_{i \in I} H_{i}
$. Display:
[
\bigplus_{i\in I} X_i = \bigtimes_{i\in J} Y_i = \bigbox_{i\in I} X_{i}= \bigcap_{i\in I} Z_i 
=  \bigcup_{n\in I} Z_{i} = \sum_{i \in I} H_{i}
]
In-line $
\bigplus_{i=0}^{\infty} X_i  = \bigtimes_{i=0}^{\infty} Y_i = \bigbox_{i=0}^{\infty} X_{i} 
= \bigcap_{i=0}^{\infty} Z_{i} =  \bigcup_{n=0}^{\infty} Z_{i} = \sum_{i =0}^{\infty} H_{i}
$. Display:
[
\bigplus_{i=0}^{\infty} X_i = \bigtimes_{i=0}^{\infty} Y_i = \bigbox_{i=0}^{\infty} X_{i} 
= \bigcap_{i=0}^{\infty} Z_i =  \bigcup_{n=0}^{\infty} Z_{i} = \sum_{i=0}^{\infty} H_{i}
]
\end{document}