Where the pickup sits under the string determines its basic tone, true, but you have it backwards. The bridge pickup is always brighter because it is near the end of the string where the high frequencies are dominant. The neck pickup is closer to the middle of the string where, as you say, the string vibrates wider -- so you get the longer waves (lower frequencies).
This is why the neck version of a particular model is almost always designed to be lower output because there is plenty of vibration for it to pickup (sometimes they're even, as with the original PAF's and Gibson's 57 Classics, etc). The bridge version is wound hotter or at least uses a hotter magnet to make up for the relative lack of vibration toward the end of the string and add some beef and low end that the bridge position lacks.