I dont' recall seeing anyone trying to answer this question, so lemme give it a try :3
If it works the way I understand it then the element system is a little misleading to be honest. As far as I've been able to discern there's two types of element: 'level' element and 'attack' element.
The level element is the 'recommended' one on the level and the one you see mention on the weapon itself. If these match up the enemies should take a little bit more damage from any/all attacks from said weapon. I've heard the difference isn't big enough to really be all that noticeable or worth caring about, however
Also, a level's element will determine which defence bonus you'll get from the badges (if any)
Attack elements are more like status effects and usually only applied via certain attacks. These can sometimes even differ from the 'level' element on the weapon (sheik can do pretty much any of them and Lana's deku stick, despite being 'water' has a few electric attacks (aka her wind blasts.)
What Midna and Fi's specials do is apply the status to ALL their attacks, where normally the majority of their attacks wouldn't apply it at all
On an extra note, it's worth checking the description for darkness+ as it doesn't seem to increase the damage but instead amplifies it to increase on multiple targets, essentially making it similar to light
If I'm wrong on any of this then lemme know but that's the general jist I've been working from based on in-game texts and a few guides :3