I know of these other techniques and used to use them but the hook is better.
The cup hook Jf1962 does move around and is not always there where you want it plus there's not much depth on the hook.
The Biner method with just a spring gate is what i used to use but you cannot just drop the saw in by holding the saw handle, you have to regrip the saw or toss it slightly in the air to grab the back strap to drop it in. Nor can you just grab your saw by the handle and go for it.
when I'm dropping big limbs off, I do two things to prevent the saw getting trapped. 1) tie the saw off to the limb just below where you are felling it, 2) if step cutting, do the 2nd cut BELOW the first, thus reducing the possibilty of the saw catching before it falls.