I disagree with the accepted answer.
The all important question is what counters what.
I think 'counter' in both 'countersink' and 'counterbore' refers to the relation between primary (often cylindrical) hole and subsequently created cone or secondary bore. The secondary cone/ bore counters the primary bore.
To see that, imagine what happens during creation of secondary cone/ bore. The image is that the secondary structure gradually grows in size, e.g. length, diameter, towards the primary by replacing or pushing back the latter, thus 'counter'.
'countersink' is essentially a verb. Thus 'countersunk hole', and by extension 'countersunk screw'.
'counterbore', in our context here, is often a noun, much like 'bore'. But 'to counterbore' is also used, for example, the progressive form 'counterboring'.
I don't think at all that 'counter' refers to the relation between hole (including any secondary form) and any insert.
To support this, note that a cylinder head screw is never called a 'counterbore screw'.
Still, it's true that the screw head has specific compatability with the (finished) hole regarding form.
Indeed, you need appropriate screw for any screw hole.
However, that's not the essence of 'to countersink' or 'counterbore'. The latter terms are about the relation between primary and secondary hole structure (or creation process), as opposed to hole-insert relation.
('countersunk screw' simply means 'screw appropriate for countersunk hole'. It is the hole itself that's created, in part, by countersinking or counterboring.)
As time goes by, I'll add further evidence to support my theory.