“Don't try the paranormal until you know what's normal.” - Granny Weatherwax
It's neither of those actually. Look at your player board, the timing is on there. You score VPs after you've finished playing cards and taking actions, and before you discard and draw a new hand.
It could have been clearer in the rulebook I think, but there's no doubt about the intended timing when the turn sequence on the player board is taken into account.
Oh! Good catch (I only looked at the rulebook). That's a bit stricter than I initially thought.
Then, I think the non-VP bonus from Site Control tokens (namely, the added influence) it's what the rulebook refers to when it mentions "getting an effect that depends on your degree of control".