I would definitely go the second way with $\displaystyle: I find that it looks strange to have something centered after a bullet. To get the vertical alignment correct, just add [t] after \begin{aligned}, where "t" means "top". (See also the top of page 8 of the User's guide for the amsmath Package.)
You can use align and (short)intertext:
\documentclass{article}
\usepackage{mathtools,lipsum}
\begin{document}
\begin{align}
\begin{array}{l@{\hspace{3mm}}c@{\hspace{3mm}}l}
... & := & ... \\[4mm]
& =: & ...
\end{array}
\intertext{\lipsum*[1]} %% or \shortintertext{\lipsum*[1]} from mathtools
\begin{array}{l@{\hspace{3mm}}c@{\hspace{3mm}}l}
... & := & ... \\[4mm]
& =: & ...
\end{array}
\end{align}
\end{document}
Dealing the edited question:
You can get rid of array too:
\documentclass{article}
\usepackage{showframe} %% just for demo
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsfonts}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{relsize}
\usepackage{stmaryrd}
\usepackage{bm}
\newcommand{\h}[1]{\widehat{\bm{#1}}}
\newcommand{\he}{\ensuremath{\tfrac{1}{2}}}
\newcommand{\quat}[2]{\mathlarger{\llbracket\,} \cos{(\he#2)}, \, \sin{(\he#2)} \,\h{#1} \mathlarger{\,\rrbracket}}
\newcommand{\quatv}[1]{\mathlarger{\llbracket\,} 0, \,\h{#1} \mathlarger{\,\rrbracket}}
\begin{document}
\begin{align}
\quatv{b} \cdot \quatv{a} & := \mathlarger{\llbracket} -\h{b}\cdot\h{a}, \, \h{b}\times\h{a} \mathlarger{\,\rrbracket} \quad \equiv \quad \mathlarger{\llbracket\,} \h{a}\cdot\h{b}, \, \h{a}\times\h{b} \mathlarger{\,\rrbracket} = \\[4mm]
& =: \quat{n}{\gamma}.\\
\intertext{In the group this means that ... }
L_b \circ L_a & =: R(\gamma \h{n}).
\end{align}
\end{document}
If this goes for long, you may add \allowdisplaybreaks in your preamble.
Best Answer
MathType is not really doing a good work: it's much simpler to type in LaTeX directly.