Why is dominating set in $W[2]$, but independent set in $W[1]$

Question

In Parameterized Complexity the Independent Set Problem for a Parameter $k$ ist $W[1]$-complete, and Dominating set is $W[2]$-complete. Now the prototypical $W[1]$ problem is deciding by a single-tape non-deterministic Turing machines if it has an accepting run with at most $k$ stepts, and for $W[2]$ it is the same notion with a multitape machine.

Intuitively they are both set selection problems, so a non-deterministic machine can guess in at most $k$ steps what elements to take into an independent (or dominating) set. So with this intuition, that the first problem ist in $W[1]$ is clear, but not why dominating set is not? So what am I missing, what is the distinguishing feature, why do we need a multitape machine for the second, but not for the first? My gut feeling would put domination set into $W[1]$, but that is not true as I know.

EDIT: I took the Turing machine definition of $W[1]$ from the following seminal paper Fixed-Parameter Tractability and Completeness I: Basic Results, but it is also written on wikipedia, together with the characterisation of $W[2]$. If you do not have access to the paper you can get a pdf from here, but the diagrams are missing in there (see page 6 for the definition of the Turing machine problem).