As I said in this answer, for these types of questions I really like Detection Theory: A User's Guide by Macmillan and Creelman. They consider 3 types of bias ($c$, $c'$, and $\beta$) that differ in how they behave when the index of sensitivity $d'$ changes, but all three of their definitions agree with your professor that a conservative observer always replies no more often than yes, regardless of the probability of a signal being present. All three definitions also lead to the conclusion that the ideal (maximum likelihood) observer is biased when the probability of a signal being present is not equal to 0.5 (with the standard Gaussian distributions with equal variance assumptions). While it may be uncomfortable to you that the ideal observer is biased, this happens all the time in estimation. That said, Macmillan and Creelman defined three different biases, there is no reason we cannot define a fourth that behaves as you want it to ...
I suggest starting with Macmillan and Creelman's definition of $\beta$. They define $\beta$ to be equal to $p(x|S_2)/p(x|S_1)$ where $p(x|S_1)$ is the probability of observing $x$ given stimulus $S_1$. Within their framework, an unbiased observer has a $\beta$ of one. If we define $\beta'$ to be $p(S_2|x)/p(S_1|x)$ and an unbiased observer as having a $\beta'$ equal to 1, then the ideal observer is unbiased. That seems like a nice result. In the case where $p(S_1)$ is equal to $p(S_2)$, $\beta$ is equal to $\beta'$, so that is good too. When $p(S_1)$ is less than $p(S_2)$ a conservative observer could say yes more than no, which I don't like, but that is the case when we force the ideal observer to be unbiased.
