Connection between two types of convergence of random variable

Question

Let $\{\xi_n\}_{n \geq 1} : (\Omega, \mathcal{F}, P) \rightarrow (\mathbb{R}^1, Bor)$ be a sequence of r.v.

Please, help me find connections between the following types of convergence as $n \rightarrow \infty$:

1. $\xi_n \xrightarrow{sLip} \xi$, i.e. $\Sigma_{n \geq 1} E|f(\xi_n) - f(\xi)| < \infty$ $\forall f \in Lip$ and bounded

2. $\xi_n \xrightarrow{sdistr} \xi$, i.e. $\Sigma_{n \geq 1} E|F_{\xi_n}(x) - F_{\xi}(x)| < \infty$ $\forall$ point $x$ of continuity of (distribution) $F_{\xi}$.

I know that in special cases (for example when $f$ is $arctan$) the second convergence follows from the first convergence. But I don't know how to prove it in general (and find a counterexample that first convergence does not follows from the second, if I understand this correctly).