Consider two metrics on $\{x,x',y\}$ defined by
$$|x-y|_1=|x'-y|_1=|x-y|_2=3, \quad |x-x'|_1=|x-x'|_2=1, \quad |x'-y|_2=2.$$
Denote by $X_1$ and $X_2$ the corresponding metric spaces.

Note that $\mathrm{pack}_\varepsilon X_1\equiv \mathrm{pack}_\varepsilon X_2$.
Indeed, for both spaces we have

 - $\mathrm{pack}_\varepsilon=1$ if $\varepsilon>3$,
 - $\mathrm{pack}_\varepsilon=2$ if $3\geqslant \varepsilon>1$, and
 - $\mathrm{pack}_\varepsilon=3$ if $1\geqslant \varepsilon>0$.

The identity map on the set $\{x,x',y\}$ defines an onto short map $X\to X'$ which is not an isometry.