Either way will work. By the way, I put more emphasis on cooling the body cavity down. The carcass can always be washed out and cleansed. I will not leave home without my Gerber Exchange-a-blade saw, one of the most valuable tools I own while hunting.

Zip saw, pelic saw, seesaw... can't say I've hunted with anyone that use them. I know an old timer that used a stout knife blade to split the pelvis, but I have never needed to perform such an act. Field dress, remove associated internals... including the bladder, and get going. Heck, even my taxidermist, who can completely disect a deer with a pen knife in under 5 minutes, (and I do mean completely), doesn't tamper with the pelvis afield. No need.

I use a sawzall to cut the pelvis after the deer has been properly aged and skinned. I won't split the pelvis before I start to cut up the meat 'cause the hide pulles away and you wind up with some dry, wasted steak and roast meat. Plus the leggs stay together and don't flop around while the deer is being hauled out, stays much cleaner.
Just the way I was taught how and it works for me.

Nope, Never cut the pelvic bone prior to hangin here either, used to but never again. By cutting that bone you stand risk of contaminating the meat around the area not to mention the pain of the back legs flopping around when dragging a deer out. I ring the anus and pull the entrails out. Keeps the meat much cleaner and makes the drag easier.