Phase factor in the equal time commutation relation in Klein-Gordon field

Two related questions regarding the equal-time commutation relation in the Klein-Gordon field (I suspect they have the same/very related answers):

In the following notes http://www-thphys.physics.ox.ac.uk/people/FabianEssler/C6web2012/theory/canonical.pdf, they have:

My question is going from the second to the third line — shouldn’t we have an overall factor of $e^{- i (omega_p – omega_q)t}$ in front of the integral because the $p, q$ in the second line are four-vectors and not three-vectors?

Alternately here http://cftp.ist.utl.pt/~gernot.eichmann/2015-qft/qft-2.pdf, they have:

and my question is why we don’t have $e^{- i (E_p – E_q)t} [a_p, a_{p’}^dagger] = [a_p(t), a_{p’}^dagger(t)]$ instead of the $[a_p, a_{p’}^dagger] = [a_p(t), a_{p’}^dagger(t)]$ shown in the equations above?