Electrode potential for metal-metal ion and metal-metal insoluble salt-salt ion electrode
My question is that $E^\circ$ is just the potential at some given standard condition, i.e. 1 bar and 25 °C....


No, $E^\circ$ is the potential at the given standard condition 25 °C, 1 ATM and activities of involved compounds equal to 1.

This may be the source of your confusion.

Standard potentials are for the unit activities. In my example, once for silver ions, once for chloride ions. 

The activity of $\ce{Ag+}$ in 1M $\ce{AgNO3}$ Is very different to activity of $\ce{Ag+}$ in 1 M $\ce{KCl}$ with the $\ce{AgCl}$ precipitate. 

You cannot find pressure in the Nernst equation. The pressure plays only very marginal role here, like being big enough to avoid water boiling at low pressure.

Usage of each of the standard potentials implies usage of activity of different ions to compute the final potential, or concentations as approximations.

One is using the metal cation activity, the other the anion activity.

One is using the metal cation activity directly, the other indirectly via the salt solubility product.

$$\begin{align}
E_\mathrm{Ag/Ag+}&amp;=E^{\circ}_\mathrm{Ag/Ag+}+0.059\log{\left(a_\mathrm{Ag+}\right)}\\
&amp;=E^{\circ}_\mathrm{Ag/Ag+}+0.059\left(\log{\left(K_\mathrm{s,AgCl}\right)}
-\log{\left(a_\mathrm{Cl^-}\right)}\right)\\
&amp;=E^{\circ}_\mathrm{Ag/AgCl/Cl^-}
-0.059\log{\left(a_\mathrm{Cl^-}\right)}\\
\end{align}$$