What's the difference between B and Exp(B) in GLM? And is it more correct to replace B with β?
I don't need an extremely detailed answer about the theory behind it.
Using the log link the coefficients are on the log scale so for ease of interpretation most people exponentiate them.
Greek letters are used for the population vales so the sample value is usually represented as $b$ not $\beta$. In fact since it is an estimate of $\beta$ you can write it as $\hat\beta$