cleveref and customized refname

Mico's answer is of course interesting but it looks like new developments are needed. Instead a manual solution is possible:

However, hyperref[eq:cos]{Condition~eqref*{eq:cos}} should be considered with care.

It is thus possible to define a new command (here for equations) with two arguments (the term to be used and the label)

newcommand*{myrefeq}[2]{hyperref[#2]{#1~(ref*{#2})}}

We can then use it with the appropriate term, depending on the need: In myrefeq{Equation}{eq:label}... or In myrefeq{Condition}{eq:label}... where the hyperlink is on the full Equation (1) or Condition (1).

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}{#2textup{(#1)}#3} % same as 'equation'

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