I think the mk1 S model has a flat plastic badge that fits into a recess in the airbag. I've just nipped out for a look at the daughter's Jazz SE, and it is exactly like the photo of the mk2 wheel you posted - the chromed emblem is set into the shaped plastic of the airbag cover.
Looking around the internet, it seems that the fixings for the badge are moulded into the cover, and are very difficult to remove without damaging the emblem. If you could get a scrap blown airbag, you could have a go at removing it by cutting around it to avoid damage... If that's successful, you could then have a go at removing the raggy one on your wheel.
It sounds like the stick on gel badges are unsatisfactory too, it appears that they peel at the edges due to the shape of the recess in the wheel.
£15 for an airbag with a good badge still sounds the best bet.