Melody Chan gave two definitions of embedded tropicalization in her Lectures on Tropical Curves and Their Moduli Spaces. Let $K$ be a nonarchimedean valued field and $X$ be a subvariety of $(\mathbb{G}_m)^n$, which in other words means $X$ is defined by the ideal of the Laurent polynomial ring $K[x_1^{\pm},\cdots,x_n^{\pm}]$. One definition is that the tropicalization $Trop(X)$ of $X$ is the subset of $\mathbb{R}^n$: $$\{(v(x_1),\cdots,v(x_n))|(x_1,\cdots,x_n)\in X(L)\, for\, L/K\, a\,valued\,extension\},$$ where a valued extension $L/K$ means $L$ is a valued field over $K$ whose valuation restricts to $v$ on $K$.
Another definition is that $Trop(X)$ is the image of the map $X^{an}\to\mathbb{R}^n$ that sends $p:Spec(L)\to X$ to $(\nu_{x_1}(p),\cdots,\nu_{x_n}(p))$, where $X^{an}$ consists of all the maps $p:Spec(L)\to X$ and $\nu_{x_i}(p)=v(p^\sharp(x_i))$. My question is how to understand $X(L)$. Usually it means the fibre product, but why are these two definitions compatible? Generally we don't have $K[x_1^{\pm},\cdots,x_n^{\pm}]/I\otimes_KL\to L$.