Cylindrical Brownian motion?

Question

There's two different notions of cylindrical Brownian motions on a Hilbert space and I can't quite link them together:

1. The first definition (for example used in Liu/Röckner SPDE'S:An introduction) of a cylindrical Brownian motion on a Hilbert space $H$ is quite intuitive, as it's just a Brownian motion which takes values in a Hilbert space $\tilde{H}$ such that the embedding $\tilde{H}$ into $H$ is Hilbert Schmidt. That is $W:[0,T]\times\Omega\rightarrow \tilde{H}$ such that:
   For $Q\in L(H)$ a $H$-valued stochastic process $(W(t))_{t\in[0,T]}$ on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ is called $Q$-Wiener process if
   (a)$W(0,\omega)=0$ for all $\omega\in\Omega$,
   (b)$W$ has a.s. continuous trajectories,
   (c) for every $0=t_0< t_1<\dots< t_n\leq T$ the increments \begin{align*} W(t_i)-W(t_{i-1}), i=1,\dots,n \end{align*} are independent and $\mathcal{N}\big(0,(t_i-t_{i-1})Q\big)$ distributed. The $Q$-Wiener process is called $H$-cylindrical B.M. if the embedding $\tilde{H}$ into ${H}$ is Hilbert-Schmidt.

2. The second definition (for example found in the notes by Jan van Nerven) is via isonormal processes, that is $W:L^2(0,T;H)\rightarrow L^2(\Omega)$ such that $W$ is an $L^2(0,T;H)$-isonormal process.

I can sort of link the 2 concepts together by condsidering an ONB $(h_n)_{n\in\mathbb{N}}$ of $H$ and the function $1_{[0,t]}\otimes h_i:[0,T]\rightarrow H$ defined by $(1_{[0,t]}\otimes h_i)(s):=(t\wedge s)h_i$. Then $W(1_{[0,t]}\otimes h_i)$ as in second definition is the $i$-th coordinate of the first definition.

Is this the link or is there a better and more intuitive connection between these concepts?