Are these two definitions of "good kernels" in fourier series equivalent?

Question

This is the definition of good kernels from the text by Stein and Shakarchi:

Now this is an alternate definition also given by many texts in Fourier analysis:

So the only difference between these two definitions is whether the kernel is positive or whether the integral of the modulus is bounded.

It is possible to show the Dirichlet kernel fails to be a good kernel under both definitions, but are these definitions completely equivalent, or is one definition stronger than the other?