I’ve been spending a lot of time trying to help people define what a push-up is to them following basic principles your videos have offered.  I look for things like a good plank; trunk stability and note if and when it breaks; proper setup and initiation through the shoulder; position in the bottom and a vertical foream etc.  We use our squat racks or boxes to slowly lower people to the ground as they get stronger and also work thing like a solid plank into an eccentric lowering with good position.