I have nothing against starting out with motivating examples, obviously they are needed for understanding.
But they should motivate the definition of a vector space. Not the definition of vectors as mappings of indices.
Functions are actually a great motivating example for the definition of a vector space, precisely because they are first look nothing like what student think of as a vector.
Thinking about this specific case, I think you are right. The manner of describing actually confuses the concept more than if it never tried to introduce the index-mapping.
Functions are actually a great motivating example for the definition of a vector space, precisely because they are first look nothing like what student think of as a vector.