I think the how depends on the equipment at hand and ground conditions.
I've never had an issue lifting posts out with front end loader, but I've always had a utility tractor and always had the chain on the bucket between the loader arms and lifted straight up.
I have also done some with 3pt, need a cross bar to do it right unless you let the 3pt arms touch and link them together with a hitch pin through the ball ends. Can't reach across a ditch too well with the 3pt either.
I tried to lift a large post out with 3pt hitch once with a 2wd utility with no loader, the front end would come off the ground as fast as the 3pt hitch lifted. That was the most unstable thing that's ever happen to me pulling posts.
For the vinyl issue, try a "choker" strap instead of a chain?