I am confused about the following notation:
For a simple case, let $X=R^d$ or $X=R$. What dose $$ \int_X \|x\|^pd\mu(x) $$ mean for a Borel probability measure $\mu$?
For $X=R$, then $x\in R$ is a 1-dimensional case. If we take $\|\cdot\|$ is the Euclidean metric. So $$ \int_X \|x\|^pd\mu(x)=\int_R \|x\|_2^pd\mu(x)=\int_R|x|^pd\mu(x)=E_{X\sim \mu} [|X|^p] $$
However, I am confused about $X=R^d$. It seems that it will be for $x=(x_1,\dots,x_d)$, $$ \int_X \|x\|_2^pd\mu(x)=\int_{R^d} (\sqrt{x_1^2+\dots+x_d^2})^pd\mu(x) $$ Is that right?
For $\mathcal X=\mathbb R^d$, you can choose the Euclidean norm or the $\ell^p$ norm, which is equivalent, that is, there exists constants $C_{p,d}$ and $C'_{p,d}$ such that $C_{p,d}\left(\sum_{k=1}^d\lvert x_k\rvert^p\right)^{1/p}\leqslant \sqrt{\sum_{k=1}^dx_k^2}\leqslant C'_{p,d}\left(\sum_{k=1}^d\lvert x_k\rvert^p\right)^{1/p}$. With this in mind, we derive that $$ \mathcal W\left(\mathbb R^d\right)=\left\{\mu\in\mathbb P\left(\mathbb R^d\right), \int_{\mathbb R}\lvert x\rvert^p d\mu_i(x)<\infty\mbox{ for each }1\leqslant i\leqslant d\right\} $$ where $\mu_i$ is the $i$-marginal: $\mu_i(B)=\mu\left(\mathbb R^{i-1}\times B\times \mathbb R^{d-i}\right)$.
You characterisation is also correct by the way.