^ This.
Rank is relatively independent of position. Squad Commander is like saying Sensors Officer or TO. So, for example, not all TO's are Prince Commanders, and not all Prince Commanders are TO's. In fact, Ossanlin was only a Prince Lieutenant during his tenure as the Galaxy Tree's TO.
Eleric doesn't want to highlight the fact that he's only a Warrior, so he's giving Efaen only the most impressive truths he can. In this case, Eleric's position is much more impressive than his rank. Of course, it can be taken two ways. Either Eleric is such a gifted pilot that he deserves the position in spite of his rank, or he's merely the best choice to fill the position but isn't mature enough to deserve a rank promotion. In Eleric's case, it's a little bit of both.
And yes, Claxter can give Eleric orders and expect to be obeyed as long as Eleric's not under any orders from an Andalite that's higher-ranked than Claxter.