As domotorp already wrote, the equivalence is easy: on the one hand, a deterministic verifier can simulate and check the computation if given the sequence of nondeterministic choices made by the NL algorithm on the witness tape, on the other hand, a nondeterministic logspace machine can simulate the verifier while replacing the witness tape with nondeterministic guesses.

If the verifier has unrestricted access to the witness tape, the class you get is all of NP. On the one hand, the logspace verifier is in particular polynomial-time. On the other hand, given a nondeterministic poly-time machine, one can use the transcript of its accepting run as a witness, and one can check the correctness of the transcript by a logspace algorithm.