documentclass[11pt,oneside,article]{memoir}
usepackage{lipsum}

% Begin Section format
newcommand{marginbox}[1]{%
  parbox[t][0pt]{10em}{%
  upshapesffamilyraggedrightmbox{}#1}%
}
newcommand{marginhead}[1]{%
  bfseriessmash{raisebox{baselineskip}{llap{marginbox{#1}kern3em}}}%
}

setsecindent{0em}
setsecheadstyle{marginhead}
setbeforesecskip{1em}
setaftersecskip{0em}
setsechook{%
  setsecnumformat{csname   the##1endcsnameenspaceparnobreak}%
}
% End Section format

copypagestyle{myheader}{ruled}
makeevenhead{myheader}{}{}{My-Name}
makeoddhead{myheader}{}{}{Other-Name}
pagestyle{myheader}

begin{document}

% Layout
settrims{0pt}{0pt}
settypeblocksize{562pt}{300pt}{*}
setulmargins{114pt}{*}{*}
setheadfoot{28pt}{24pt}
setheaderspaces{*}{*}{*}
setmarginnotes{1pt}{1pt}{1pt}
setlrmargins{210pt}{*}{*}
checkandfixthelayout

section{Introduction}
lipsum[4]

end{document}

documentclass[11pt,oneside,article]{memoir}
usepackage{lipsum}

% Begin Section format
newcommand{marginbox}[1]{%
  parbox[t][0pt]{10em}{%
  upshapesffamilyraggedrightmbox{}#1}%
}
newcommand{marginhead}[1]{%
  bfseriessmash{raisebox{baselineskip}{llap{marginbox{#1}kern3em}}}%
}

setsecindent{0em}
setsecheadstyle{marginhead}
setbeforesecskip{1em}
setaftersecskip{0em}
setsechook{%
  setsecnumformat{csname   the##1endcsnameenspaceparnobreak}%
}
% End Section format

copypagestyle{myheader}{ruled}
makeevenhead{myheader}{}{}{My-Name}
makeoddhead{myheader}{}{}{Other-Name}
pagestyle{myheader}

%%% Fix sectionmark
makeatletter
renewcommandsectionmark[1]{%
  @setclcnt{section}{@memmarkcntra}%<------ Missing!
  advancec@@memmarkcntram@ne
  markright{%
    memUChead{%
      ifnumc@secnumdepth>c@@memmarkcntra
        if@mainmatter
          @nameuse{sectionmarksn}{@nameuse{thesection}. space}%
        fi
      fi #1%
    }%
  }%
}
makeatother

% Layout (should go before begin{document})
settrims{0pt}{0pt}
settypeblocksize{562pt}{300pt}{*}
setulmargins{114pt}{*}{*}
setheadfoot{28pt}{24pt}
setheaderspaces{*}{*}{*}
setmarginnotes{1pt}{1pt}{1pt}
setlrmargins{210pt}{*}{*}
checkandfixthelayout

begin{document}


section{Introduction}
lipsum[4]

end{document}