As a matter of fact - here (A320neo, A321neo, B737MAX-8) we see those differ a LOT. With A350 and B788 they seem to be the same.
Does someone remember how it was with other models ?
Very often there are more than one launch customer. There were several of them, for instance, when the 787-10, the 778/9 and the A330neo were launched. As usually one of them also is the first operator, we tend to remember it as 'the launch customer'.
Launch customer is a status, typically more than one airline is needed to sign an order contract to turn a concept into a program because of proven real world demand. Launch customers get rebates and can influence many details like the first version of a family available or a certain engine being certified first. Aside from that other airlines are participating in early stages with quite significant input without being launch or just customers, just in case they want to order it at a later time.