The split ones are made so that you don't have to remove the driveshaft, however, in my experience the slide on ones are much stronger in the long run as they're all in one piece. If your Mechanic has asked for the ones you have shown us, they're the best ones to use. The glue on versions are also more fiddly to fit, too.
The stretch ones are fitted by removing the driveshaft assembly etc and using a cone to ease the boot on, inside-out. You the simply turn it the right way round and it should fit nicely. Grease is supplied in the packet with the boot, and it's a good idea to use boiling water to relax the rubber so it expands easier. If you can picture the diametre of the driveshaft, then the small end of the new boot, you'll be able to see just why you need the hot water, lol.