let $X,Y,Z$ be random variables defined on the same probability space and let covariance of $X$ and $Y$ be finite, then the law of total covariance / covariance decomposition formula states: \begin{align} Cov(X,Y)=\underbrace{\mathbb{E}\big[\text{Cov}(X,Y\lvert Z)\big]}_{\text{(i)}}+\underbrace{\text{Cov}\big[\mathbb{E}(X\lvert Z),\mathbb{E}(Y\lvert Z)\big]}_{\text{(ii)}} \end{align} What is the interpretation of $\text{(i)}$ and $\text{(ii)}$?

My thoughts: in (ii) the two conditional expectations can be seen as random variables themselves, I also know that this is a generalization of the law of total variance / variance decomposition formula which can be shown by setting $X=Y$, where the interpretation is then that of a variation in $Y$ explained by $Z$ and unexplained by $Z$. But what is the correct interpretation in the above covariance formula for (i) and (ii)? Wikipedia offers a brief description which is not very satisfying.

let $X,Y,Z$ be random variables defined on the same probability space and let covariance of $X$ and $Y$ be finite, then the law of total covariance / covariance decomposition formula states: \begin{align} \text{Cov}(X,Y)=\underbrace{\mathbb{E}\big[\text{Cov}(X,Y\lvert Z)\big]}_{\text{(i)}}+\underbrace{\text{Cov}\big[\mathbb{E}(X\lvert Z),\mathbb{E}(Y\lvert Z)\big]}_{\text{(ii)}} \end{align} What is the interpretation of $\text{(i)}$ and $\text{(ii)}$?

My thoughts: in (ii) the two conditional expectations can be seen as random variables themselves, I also know that this is a generalization of the law of total variance / variance decomposition formula which can be shown by setting $X=Y$, where the interpretation is then that of a variation in $Y$ explained by $Z$ and unexplained by $Z$. But what is the correct interpretation in the above covariance formula for (i) and (ii)? Wikipedia offers a brief description which is not very satisfying.