Logarithmic space verifier with unbounded witness
Let $N$ be the total number of configurations of the machine other than the witness tape, namely state, location of the input tape head, contents of the work tape, and location of the work tape head. Note that $N$ is polynomial in $n$.

We can assume without of generality that at each step, the machine reads a bit from the witness tape, and it affects its decision (we can accommodate that by adding dummy bits to the witness tape at locations where the original machine doesn't read the witness tape). The machine terminates once the witness tape runs out.

I claim that for any witness $w$, there is an equivalent witness $w'$ with $|w'| &lt; N$. Here equivalent means that when the machine terminates, it is in the same configuration. The proof resembles that of the pumping lemma: if $|w| \geq N$, then some configuration must repeat, and so we can remove the corresponding part from the witness.

Thus, without loss of generality, the witness is of size at most $N=\operatorname{poly}(n)$. So this definition coincides with the usual definition of $\mathsf{NL}$.