Conflict of domain and endpoints in Noether's theorem and energy conservation

In the derivation of energy conservation, there is the transformation $$q(t)\rightarrow q'(t)=q(t+\epsilon)$$, whose end points are kind of fuzzy. The original path $$q(t)$$ is only defined from $$t_1$$ to $$t_2$$. If this transformation rule is imposed, $$q'(t_2-\epsilon)=q(t_2)$$ to $$q'(t_2)=q(t_2+\epsilon)$$ is not defined in the original path. Then how could the Lagrangian be integrated?

On P.98 of Jakob Schwichtenberg's book Physics from Symmetry, he stated that $$\delta q(t_1)=\delta q(t_2)=0$$ whereas Kleinert stated in his Particles and Quantum Fields $$\delta q_s(t_a)$$ and $$\delta q_s(t_b)$$ are not necessarily $$0$$. Who's correct?

This question is different from the endpoint questions since it is already clear that $$q(t_2+\epsilon)\neq q(t_2)$$.