cleveref and customized refname

You asked,

Is it possible with cleveref to customize on the fly the name in the clickable link?

Short answer: Yes. Longer answer: Your typographic objective may be achieved using cleveref's aliasing capabilities. For more information, see section 6 of the package's user guide, entitled "Overriding the Cross-Reference Type".

\documentclass{article}
\usepackage[colorlinks=true,allcolors=red]{hyperref}
\usepackage[nameinlink,noabbrev]{cleveref}

% introduce an alias for 'equation'
\crefalias{condition}{equation}
\crefname{condition}{condition}{conditions}
\Crefname{condition}{Condition}{Conditions}
\creflabelformat{condition}{#2\textup{(#1)}#3} % same as 'equation'

\begin{document}
It is true that
\begin{equation} \label[condition]{eq:cos} % note the optional argument of '\label'
\cos\pi=-1
\end{equation}
However, \Cref{eq:cos} should be considered with care.
\end{document}