Stay away from stamps and logos !As George says. Before I bring the hot iron to the briar I apply a drop of water to the sore spot.
As I do when steaming out a dent from a wooden table top, - floor etc.
Perhaps the drop of water is only superstition, but it always worked fine for me
These look more like scratches to me, but it is worth trying.
Be sure the steam function of the iron is turned off !
Careful not to burn your fingers and practice patience.
Works fine to remove toothmark from stems too. If so use cloth, not paper.
Please show us the outcome