I could be wrong buuuut ...
There are different extents of circumcision, you can have a combination of high/low (a measure of how close the scar will be to the head of the penis) and tight/loose (a measure of exactly how much foreskin is removed). A tight circumcision would leave no foreskin and the penis would be totally smooth from base to head even without an erection and a loose one would still have potentially all the foreskin and enough to cover the head.
In both cases the frenulum, the tight bit joining the underside of the foreskin and the head is almost always removed.
I did my research when I went through a circumcision phase
