I have pushed a lot of trees, both as the faller and operator.
The only time you need to cut under the face cut is if there is bad limb lock or you are trying to force the tree through vines, other trees or something similar. Remember that if your back cut is lower it will help to prevent the machine from pushing the hinge off but it also requires a lot more pressure to move the tree forward, sometimes it might not be worth the trade. If you have to push so hard so low that you are worried about the hinge you might want to consider other options. I will usually just leave a thick hinge and have the operator move the tree very slowly and advance my back cut as needed to prevent barberchair
The biggest thing is getting the operator to understand what you need. Most of the time I cut a push tree like a normal pull tree. Rest the machine against it while making face cut, add some pressure, start back cut, make sure the operator doesn't push faster than you cut, leave a good hinge, step away and signal operator to tip it over.
The operators tend to want to push to much to fast and ride the tree to the ground. Make sure that they understand that you only need them to hold the tree up and then tip it over.
My biggest scare came when falling some trees for my dad when he was adding a swimming pool. It was my dad, his skid steer, his tree and his house. He has never really respected what I do or taken directions from me very well. He was pushing a 36" loblolly pine with the skid steer. He pushed faster than I could cut, as the tree started to move and I was racing to finish my hinge I felt my leg being pushed very hard. It was the front tire of the skid steer. I ran away and let it fall with a large triangle shaped hinge. the only thing that saved my leg was the other front tire rode up the swell of the stump when he surged forward and left enough room under the the tire for me to jerk my leg out. My pants had tire tracks up to my thigh.