two different \mathcal{H}

The H comes from the standard font cmsy10, but an old version. There was a change in 2009. (The cited typesetting-script.pdf was created at 2009-02-11, using the old symbol font.)

The symbol with cmsy10.pfb from an old TeX Live 5 (2000):

The font file cmsy10.pfb from TL 5 (2000) contains:

CMSY10 1.0
%%CreationDate: 1991 Aug 15 07:20:57
% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

The current font file (2018) contains:

%%Title: CMSY10
%Version: 003.002
%%CreationDate: Mon Jul 13 16:17:00 2009
%%Creator: David M. Jones
%Copyright: Copyright (c) 1997, 2009 American Mathematical Society

Very likely there was a font change in 2009 or before that changed the calligraphic uppercase H.

I hacked a fit to 10pt and 12pt versions, by stacking white boxes over the left arm of the H.

\documentclass{article}
\usepackage{rotating,xcolor,stackengine}
\newcommand\specialHten{\mkern-3mu\rotatebox[origin=c]{-15.5}{%
\stackinset{l}{1.7pt}{c}{-.9pt}{\color{white}\rule{1.5pt}{1pt}}{%
  \stackinset{l}{1.875pt}{c}{-.81pt}{\color{white}\rule{1.5pt}{.65pt}}{%
    \rotatebox[origin=c]{15.5}{$\mathcal{H}$}}}}\mkern-3mu
}
\newcommand\specialHtwelve{\mkern-3mu\rotatebox[origin=c]{-15.5}{%
\stackinset{l}{1.9pt}{c}{-1.1pt}{\color{white}\rule{1.5pt}{1pt}}{%
  \stackinset{l}{2.53pt}{c}{-1pt}{\color{white}\rule{1.5pt}{.85pt}}{%
    \rotatebox[origin=c]{15.5}{$\mathcal{H}$}}}}\mkern-3mu
}
\begin{document}
$\mathcal{H}\specialHten\mathcal{H}$

\fontsize{12pt}{12pt}\selectfont
$\mathcal{H}\specialHtwelve\mathcal{H}$

\end{document}

If those boxes were visible, they would look like this:

Here is a version that gets you something similar using the modern toolchain (unicode-math in XeLaTeX or LuaLaTeX). A second script alphabet is available in Asana Math as a stylistic alternative.

\documentclass[varwidth, preview]{standalone}
\usepackage{unicode-math}

\setmainfont{Palatino Linotype}[Ligatures={Common,Discretionary,TeX}]
\setmathfont[Scale=MatchUppercase]{Asana Math}
\setmathfont[range={cal, bfcal}, Scale=MatchUppercase, Alternate]{Asana Math}

\begin{document}
\( \mathcal{H} \)
\end{document}

You can see a pretty comprenehsive list of the legacy calligraphic alphabets in the documentation for mathalpha (formerly mathalfa).

You might try pxtx or see if there’s another version there you like better:

\documentclass[varwidth, preview]{standalone}
\usepackage[cal=pxtx]{mathalfa}

\begin{document}
\( \mathcal{H} \)
\end{document}

Both of these are based on pxfonts/txfonts by Young Ryu.