According to your book, yours is the correct way, as our machines have gap beds.
But when you look at it, your apron is a lot safer, not only for the burnt hands thing but also the way you approach machining on the lathe.
For me to work comfortably, I will be at 90 degrees to the job, looking straight at it, and if anything is going to fly, it will be in that direction, whereas with yours, you will most probably be standing more to the rear of the machine, looking over the top of the toolpost, so out of the way of the 'shrapnel'.
I still think yours is the most civilised one.
John