Better way to remove extra space around the quote

The good solution is to use the quoting-package.

\usepackage[noorphans,vskip=0.5ex]{quoting}

Change all your

\begin{quote}
[citation text]
\end{quote} 

to

\begin{quoting}
[citation text]
\end{quoting}

I suggest you read the manual to learn the other useful options.

Without packages, you can redefine quote to issue zero \topsep.

\documentclass{article}
\usepackage{lipsum}

\renewcommand{\quote}{\list{}{\rightmargin=\leftmargin\topsep=0pt}\item\relax}

\begin{document}

\lipsum*[2]
\begin{quote}
\lipsum*[3]
\end{quote}
\lipsum[4]

\end{document}

Maybe you want to set the topsep to another value.

Below lines worked for all quotes.

\usepackage{etoolbox}
\AtBeginEnvironment{quote}{\vspace{-\baselineskip}} % before block quote
\AtEndEnvironment{quote}{\vspace{-\baselineskip}} % after block qoute