Saeviomagy
Adventurer
No, they later go on to affect a target that you hit with an attack. The text has to be referring to the target of the attack, not the target of the spell - otherwise the spells targeting restriction of 'self' would mean you damage and banish yourself with banishing smite...It shares spells that target only you. While smites have a range of self, they later go on to target what you hit.
Also, compare with divine favour. Is the difference in meaning between "the target of an attack takes..." and "your attacks deal" really that big? Or would you stop divine favour from working with this too?