I agree with this and what Aida said about focus/channeling. I think desire is the main element, and that's why spells require practice; you have to learn to focus on what you want it to do properly. It's just a lot easier with both spells and wands than without either (or both).
I think that's a very interesting idea. Like you said, the way magic works is not ever explained. Maybe each culture has a different idea of how magic flows. In the books they had to practice the spells, they didn't just say the words and poof! it works. Actually, your idea reminds me of Apparition- focusing on the location. To me that seems like channeling your magic to bring you to the desired destination, like channeling your magic to do the desired spell. Also, when Snape taught the N.E.W.T DADA class non-verbal spells he told them to focus on the spell they wanted to cast it since they couldn't say the incantation.