Do alleles, by definition, code for the same trait?
I fear you won't get a definitive answer on this. A survey indicates both usages of trait all over the Internet. As a teacher, I generally used trait to apply to the gene. This made teaching the "central dogma" clear: DNA -> RNA -> protein -> trait. Of course this simplification gets added to later as the course progresses. 

One interesting question this poses is in the case of skin color. Is that a trait? Last I read, at least four different genes interact to produce skin color.

I'd side with you on this but acknowledge the fact trait doesn't seem to have the consistent, well-defined meaning that allele does.

On the other hand, a visit to UC Berkeley's evolution site shows them applying trait to the variations due to allele combinations.

Hope that helps! :)