When cutting off a big branch I do it in 3 stages so it doesn't peel off the bark when it starts falling down;
I first cut into the branch from underneath making sure not to cut all the way through but only 1/3. As you can see I start cutting away from the left branch ...
... then I move the saw 10 cm upwards from the undercut and cut off the branch entirely (make sure it doesnt bounce at you and that no one is under the tree when it falls). The falling branch will only peel off the bark to the undercut line.
The rest of the branch is not heavy enough to peel off much of the bark and one can hold it with the other hand so it doesnt peel off anything. Note how close I cut the big branch off from the remaining thiner branch. The less of the old branch one leaves the better the tree will heal that cut point. If that old branch is cut away from the remaining branch the tree will get infected with rot.
All (but one) apple trees are infected with Apple Tree Rot and most of their branches have these cancer knots which I tried to cut off.
... making sure not to fall down. 4 Apple trees finished 3 to go.