Pseudorandom generators indistinguishable by uniform deterministic adversaries
Yes, this notion has been studied. One interesting aspect is that the two notions of pseudorandomness known to be equivalent under the usual adversaries, "next bit predictability" and "indistinguishability", do not seem to be equivalent for deterministic adversaries. (If they were, we would have complexity class separations.) Here are three references; I'm sure there are more.

O. Goldreich and A. Wigderson. On pseudorandomness with respect to deterministic observers

O. Goldreich. In a world of P=BPP

R. Williams. Towards NEXP versus BPP?