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.
Here are a couple episodes that hit push up positioning
Movement Transfer Exercise: The Pushup & The Slingshot
Episode 34: Bridal MWod: Bride/Operator Shoulders; IR Rotation/Scap Gliding