Poisson deviance simplifies when an intercept is fitted ($D=2{\sum }_i{y}_i\log (y_i/e_i)$)

The full Poisson deviance is $2{\sum }_i[y_i\log (y_i/e_i)-y_i+e_i]$. An unpenalized log-mean intercept has likelihood score ${\sum }_i(y_i-e_i)$, which vanishes at an interior fit. The linear terms then cancel, giving the displayed expression. Without that intercept score equation, discarding the linear terms is not justified. Use the limiting convention $0\log (0/e_i)=0$.