I would say that it only applies to enemies brought in under the Enmity Clause because this is essentially a twist brought on by the failed roll. You stated enemy on the other hand is created during PC creation and I don’t see why they should get a mechanical advanatage against you in conflicts.
For whatever it’s worth I disagree. I would think the +3s most definitely applies to the enemy a player names at character creation. An enemy’s an enemy whether he is brought in at the beginning of play or through the Enmity Clause. But I don’t have anything in the book to back this up; it’s merely my reading of the rule. Your interpretation is certainly supported by the fact that this rule is under the Enmity Clause section. I would just choose to apply it in all cases.