It is not the case that one copula density has random variables as arguments and the other does not. A copula density is just a joint PDF where the underlying random variables have uniform marginal distributions.
In general, for a probability space $\Omega$ and random variables $Y_1: \Omega \to [0,1]$ and $Y_2: \Omega \to [0,1]$ the joint PDF $f$ is a function mapping the range $[0,1]\times[0,1]$ of $(Y_1,Y_2)$ into $\mathbb{R}$, that is $f:[0,1] \times [0,1] \to \mathbb{R}$, such that for real numbers $y_1,y_2 \in [0,1]$
$$F(y_1,y_2) =\mathbb{P}(\{\omega \in \Omega | Y_1(\omega) \leqslant y_1\} \cap \{\omega \in \Omega | Y_2(\omega) \leqslant y_2\}) = \int_0^{y_1} \int_0^{y_2} f(t_1,t_2)\, dt_1 \, dt_2$$
With regard to the copula density $c(u_1,u_2)$, the arguments $u_1$ and $u_2$ are always real numbers in $[0,1]$. This is a joint PDF for random variables $U_1$ and $U_2$ with uniform marginal distributions.
What may have been confusing is that the notation for the Gaussian copula does not show the explicit dependence on $u_1$ and $u_2$ on the RHS. The inverse of the normal CDF $\Phi^{-1}$ is a mapping from $[0,1]$ to $\mathbb{R}$ and the intermediate variables $x_1$ and $x_2$ are images $x_1 = \Phi^{-1}(u_1)$ and $x_2 = \Phi^{-1}(u_2)$.
The Gaussian copula density can be written more clearly as
$$c\left(u_{1}, u_{2}\right)=\frac{1}{\sqrt{1-\rho_{12}^{2}}} \exp \left\{-\frac{\rho_{12}^{2}\left([\Phi^{-1}(u_1)]^{2}+[\Phi^{-1}(u_2)]^{2}\right)-2 \rho_{12} \Phi^{-1}(u_1)\Phi^{-1}(u_2)}{2\left(1-\rho_{12}^{2}\right)}\right\}$$
A two-dimensional copula is a function $C:[0,1] \times [0,1] \to [0,1]$ with the following properties. For all $u_1,u_2 \in [0,1]$,
$$C(u_1,0) = 0, \,C(0,u_2) = 0,\\C(u_1,1) = u_1, \,C(1,u_2) = u_2,$$
and for all $0 \leqslant u_1 \leqslant u_1' \leqslant 1$ and $0 \leqslant u_2 \leqslant u_2' \leqslant 1$,
$$C(u_1',u_2') - C(u_1',u_2) - C(u_1,u_2') + C(u_1,u_2) \geqslant 0$$
Archimedean copulas are a just general class of such functions -- with no relationship to the normal distribution function $\Phi$.
In general, suppose we have random variables $X$ and $Y$ with joint CDF $F_{XY}$ and marginal CDFs $F_X$ and $F_Y$. we can obtain the copula associated with this joint distribution as the function $C: [0,1] \times [0,1] \to [0,1]$ given by the composition
$$\tag{*}C(u_1,u_2) = F_{XY}(F_X^{-1}(u_1),F_Y^{-1}(u_2))$$
As long as we start with a valid joint CDF, the function defined by (*) will always meet the requirements of a copula specified above.
A special case is the bivariate Gaussian copula. Letting $\Phi$ denote the univariate normal CDF and $\Phi_2$ denote the bivariate normal CDF, the bivariate Gaussian copula is given by
$$C(u_1,u_2) = \Phi_2(\Phi^{-1}(u_1), \Phi^{-1}(u_2))$$
Best Answer
Note that with standard normal marginals
$$\Sigma=\left[\begin{array}{cc} 1 & \rho \\ \rho & 1 \end{array}\right],\,\, |\Sigma| = 1 - \rho^2$$
and
$$\Sigma^{-1}= \frac{1}{1- \rho^2}\left[\begin{array}{cc} 1 & -\rho \\ -\rho & 1 \end{array}\right], \,\, \Sigma^{-1}-I= \frac{1}{1- \rho^2}\left[\begin{array}{cc} \rho^2 & -\rho \\ -\rho & \rho^2 \end{array}\right]$$
Hence,
$$- \frac{1}{2}\mathbf{x}^{\top}(\Sigma^{-1}-I)\mathbf{x} = \frac{-1}{2(1- \rho^2)}\left[\begin{array}{cc} x_1 & x_2 \end{array}\right]\left[\begin{array}{cc} \rho^2 & -\rho \\ -\rho & \rho^2 \end{array}\right]\left[\begin{array}{cc} x_1 \\ x_2 \end{array}\right] \\= \frac{-1}{2(1- \rho^2)}\left[\begin{array}{cc} x_1 & x_2 \end{array}\right]\left[\begin{array}{cc} \rho^2x_1 -\rho x_2 \\ -\rho x_1 + \rho^2 x_2 \end{array}\right] \\= -\frac{\rho^2 (x_1^2 +x_2^2)- 2\rho x_1 x_2 }{2(1-\rho^2)},$$
and, thus,
$$|\Sigma|^{-\frac{1}{2}}\exp\!\left(-\frac{1}{2}\mathbf{x}^{\top}(\Sigma^{-1}-I)\mathbf{x}\right) = \frac{1}{\sqrt{1- \rho^2}} \exp \left\{-\frac{\rho^2 (x_1^2 +x_2^2)- 2\rho x_1 x_2 }{2(1-\rho^2)} \right\}$$