That is an alligator snapper, different species (assuming the OP is talking about common snappers)
Big common snappers should NEVER be held by the tail. They are tricky sometimes. I highly recommend gloves to guard against the claws, they're not as vicious as the beak but still capable of sending you to get some stitches.
First things first. Guage the turtle's "pissiness". If it's pissy and going to bite for sure, yeah always hold by the back of the shell. On really big snappers, there is no "safe" way to hold them that is also safe for the turtle, unless you're a ridiculously big person. I couldn't hold a 40lb snapper by the back of the shell.
Here's me holding a very big but calm wild snapper. This was taken last june in Ontario, I doubt he'd ever seen a human before. You can see what I mean by the claws. These guys are accomplished diggers (laying eggs) and can do damage to un-protected hands.
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