Connections between two constructions of infinite dimensional Gaussian measures

I think that what you are looking for is the link between the white noise measure $\mu_C$ and the isonormal process indexed by $\ell^2(\mathbb{Z}^d)$ with covariance structure given by $C$. The white noise measure $\mu_C$ is a Gaussian measure on $s'$ so that for all $\varphi \in s$, $\langle ;\varphi\rangle_{s',s}$ is a centered Gaussian random variable with variance $\langle \varphi ; C \varphi\rangle$. By an approximation argument, you should be able to give some sense to $\langle ; f\rangle$ with $f \in \ell^2(\mathbb{Z}^d)$ so that it is a centered Gaussian random variable under $\mu_C$ with variance $\langle f;C f\rangle$. Now, your second construction gives rise to a Gaussian stochastic process indexed by $\mathbb{Z}^d$ with covariance structure given $C$. By re-indexing, each element $X_j$ of this Gaussian stochastic process admits the representation $\nu_{C}(e_j)$ where $e_j=(0,\dots,0,1,0,\dots)$. Now, again by approximation, you can extend $\nu_C$ to all $\ell^2(\mathbb{Z}^d)$ and it is completely defined, for all $f,g \in \ell^2(\mathbb{Z}^d)$, by $$ \mathbb{E}\left(\nu_{C}(f)\nu_{C}(g)\right)= \langle f;Cg\rangle ,$$ and $\mathbb{E}(\nu_C(f))=0$. Now, the link is clear and you have the following equality in law under $\mu_C$, for all $f \in \ell^2(\mathbb{Z}^d)$ $$\nu_c(f) = \langle ; f\rangle.$$ This is completely similar to the classical construction of the white noise probability measure on the space of tempered distributions on $\mathbb{R}$ ($S'(\mathbb{R})$) and the classical isonormal Gaussian process indexed by $L^2(\mathbb{R})$.

The source of the confusion is not saying explicitly what are the sets and $\sigma$-algebras the measures are supposed to be on. For example, a sentence like ''By Kolmogorov's Extension Theorem, there exists a Gaussian measure $\nu_C$ with covariance $C$ on $l^2(\mathbb{Z}^d)$ which is compatible with $\mu_\Lambda$ for every finite $\mu_\Lambda$.'' is asking for trouble because it seems to say the measure $\nu_C$ is on the set $l^2(\mathbb{Z}^d)$, which is false.

Let's go back to basics. A measurable space $(\Omega,\mathcal{F})$ is a set $\Omega$ equipped with a $\sigma$-algebra $\mathcal{F}$. A measure $\mu$ on the measurable space $(\Omega,\mathcal{F})$ is a map from $\mathcal{F}$ to $[0,\infty]$ satisfying the usual axioms. From now on I will only talk about probability measures. For best behavior, the $\Omega$ should be a (nice) topological space and $\mathcal{F}$ should be the Borel $\sigma$-algebra for that topology. Suppose one has two topological spaces $X,Y$ and a continuous injective map $\tau:X\rightarrow Y$. Then if $\mu$ is a measure on $(X,\mathcal{B}_X)$ where $\mathcal{B}_X$ is the Borel $\sigma$-algebra of $X$, then one can construct the direct image/push forward measure $\tau_{\ast}\mu$ on $(Y,\mathcal{B}_Y)$ by letting $$ \forall B\in\mathcal{B}_{Y},\ (\tau_{\ast}\mu)(B):=\mu(\tau^{-1}(B))\ . $$ This is well defined because a continuous map like $\tau$ is also $(\mathcal{B}_X,\mathcal{B}_Y)$-measurable. Technically speaking $\mu$ and $\tau_{\ast}\mu$ are different measures because they are on different spaces. However, one could argue that they are morally the same. For example, one might be given the measure $\tau_{\ast}\mu$ without knwing that it is of that form, and only later realize that it is and thus lives on the smaller set $\tau(X)$ inside $Y$.

The first construction:

Let $s(\mathbb{Z}^d)$ be the subset of $\mathbb{R}^{\mathbb{Z}^d}$ made of multi-sequences of fast decay $f=(f_x)_{x\in\mathbb{Z}^d}$, i.e., the ones for which $$ \forall k\in\mathbb{N}, ||f||_k:=\sup_{x\in\mathbb{Z}^d}\langle x\rangle^k|f_x|\ <\infty $$ where $\langle x\rangle=\sqrt{1+x_1^2+\cdots+x_d^2}$. Put on the vector space $s(\mathbb{Z}^d)$ the locally convex topology defined by the collection of seminorms $||\cdot||_k$, $k\ge 0$. The strong dual can be concretely realized as the space $s'(\mathbb{Z}^d)$ of multi-sequences of temperate growth. Namely, $s'(\mathbb{Z}^d)$ is the subset of $\mathbb{R}^{\mathbb{Z}^d}$ made of discrete fields $\phi=(\phi_x)_{x\in\mathbb{Z}^d}$ such that $$ \exists k\in\mathbb{N},\exists K\ge 0,\forall x\in\mathbb{Z}^d,\ |\phi_x|\le K\langle x\rangle^k\ . $$ The vector space $s'(\mathbb{Z}^d)$ is given the locally convex topology generated by the seminorms $||\phi||_{\rho}=\sum_{x\in\mathbb{Z}^d}\rho_x\ |\phi_x|$ where $\rho$ ranges over elements of $s(\mathbb{Z}^d)$ with nonnegative values.

The measure $\mu_C$ obtained via the Bochner-Minlos Theorem is a measure on $X=s'(\mathbb{Z}^d)$ with its Borel $\sigma$-algebra $\mathcal{B}_X$.

The second construction:

Let $s_0(\mathbb{Z}^d)$ be the subset of $\mathbb{R}^{\mathbb{Z}^d}$ made of multi-sequences of finite support $f=(f_x)_{x\in\mathbb{Z}^d}$, i.e., the ones for which $f_x=0$ outside some finite set $\Lambda\subset\mathbb{Z}^d$. Put on the vector space $s_0(\mathbb{Z}^d)$ the finest locally convex topology. Namely, this is the locally convex topology generated by the collection of all seminorms on $s_0(\mathbb{Z}^d)$ . Note that $s_0(\mathbb{Z}^d)\simeq \oplus_{x\in\mathbb{Z}^d}\mathbb{R}$. Let $s'_0(\mathbb{Z}^d)$ be the strong topological dual realized concretely as $\mathbb{R}^{\mathbb{Z}^d}$. One can also define the topology by the seminorms $||\phi||_{\rho}=\sum_{x\in\mathbb{Z}^d}\rho_x\ |\phi_x|$ where $\rho$ ranges over elements of $s_0(\mathbb{Z}^d)$ with nonnegative values. However, this is the same as the product topology for $s'_0(\mathbb{Z}^d)=\prod_{x\in\mathbb{Z}^d}\mathbb{R}$.

The measure $\nu_C$ constructed via the Daniell-Kolmogorov Extension Theorem is a measure on $Y=s'_0(\mathbb{Z}^d)=\mathbb{R}^{\mathbb{Z}^d}$ with its Borel $\sigma$-algebra for the product topology a.k.a strong dual topology.

The precise relation between the two measures:

We simply have $\nu_C=\tau_{\ast}\mu_C$ where $\tau$ is the continuous canonical injection due to $X=s'(\mathbb{Z}^d)$ being a subset of $Y=\mathbb{R}^{\mathbb{Z}^d}$.