Probably "Asbestolux", Moses. Its use was banned in the early 80s. Although a hazard due to its asbestos content, it's not one of the worst types.
Yes, specialist firms will turn up with space-age suits and mobile showers etc. They'll make a big deal of it. It's their business. Gross over-reaction of course. With suitable clothing and precautions, it's quite possible to remove it safely.
Otherwise, certainly seal it up. it really is quite stable if undisturbed. I would incorporate a "vapour check" between the old and new ceilings. (Ordinary polythene.)
The only tricky part will be fixing new joists without knocking the old board around. I would suggest fixing joists to the walls, independent of the existing ceiling.
What are the rough dimensions of the shed?