overhead press to the front would be more correct. front press would be what we know as the bench press.
hence, "military press" is quite ok as a description.
I would adjust that to 'overhead front press', but OK, I get what you mean. 'Bench press' sounds to generic, adding 'flat', 'incline' or 'decline' & 'barbell/dumbbell/machine' is more precise.
'military press' says nothing about the direction of the movement.