In https://encyclopediaofmath.org/wiki/Gronwall_lemma the various forms of the Grönwall lemma in integral form are stated for NON NEGATIVE function $\phi$
And this coincides with what is written my lecture notes. However the proof we made did not used the fact that u(t) (that is $\phi$ in the above article) should be positive, so I am suspicious it is not required. Still, we used the Gronwall lemma to prove the continuous dependence of the solution of a Cauchy problem on the initial value, but in that case the u(t) that we applied the lemma to was a norm of some function so it was clearly nonnegative, but still this doesn't say if it works for variable-sign functions
In fact the wikipedia article https://en.wikipedia.org/wiki/Gr%C3%B6nwall%27s_inequality#Integral_form_for_continuous_functions highlights no assumptions are made on the sign of the function u(t) and so does not use it in the proof.
So, what is the correct version? Can the sign of u(t) be variable? If you have an example were the lemma is applied to a variable-sign function that may help, too. Do you have any reference of a proof requiring non negative u(t) as hypothesis and where that fact is clearly used ?