Thanks.
Modern speakers with toroidal magnets are like you describe. Older ones from the sixties as in the picture, are built different, as the ones on the link.
The one on the picture has a foil glued to the side openings to prevent dust reaching the coil, which is completely exposed by removing the foil. Perhaps they are named 'old alnico' type. Will find one, take a picture and post when I can.
Edited, added:
The coil is exposed by the magnet area, not on top by the cone/spider.
Resourcing to a patent drawing as the picture may take too long:
(Magnifiable)---->
http://patentimages.storage.googleapis.com/pages/US3134057-0.png
Being 1 the cylindrical Alnico magnet
2 and 3 polepieces,
4 the gap, 7 the coil, 9 the 'spider'
The coil/bobbin is exposed, visible, you can see it moving while operating..