On the degree of an ODE

Question

In many texts I find the definition of the degree of an ODE saying:

1. the equation must be polynomial in the derivatives,

2. the degree is that of the highest order derivatives.

My question is if in 1., the function itself counts as a derivative (namely the zero$^{th}$ derivative).

E.g. $y''+e^{y'}=0$ is clearly not polynomial. But what about $y''+e^y=0$ ?

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I am not asking about the order, and I know that rewriting the equation to polynomial form, if possible, is allowed.