Without knowing what rules non-native speakers are told, I would say that
Saber --> to know. to have the knowledge (Realize that even if you know how to do something you may not have the capacity or authorization to do it)
saber can also mean to taste like "esto sabe a fresa" but that is not the use we want to discuss).
Poder --> to be able or have the capacity/authorization to do something (for obvious reasons most of the time this might imply that you have the knowledge on how to do it)
I understand that the confusion might come from the difference between having the knowledge on how to do something (saber) and the capacity to do it (poder).
Poder in a positive context when you have the authority, capacity or power to do something. It can imply or not that you also have the knowledge to do it.
Poder can be used with the connotation of being able to do something (because I know how to do it and I'm authorized to do it) but
Saber usually only implies the knowledge. Since sometimes having the knowledge may be enough,
Poder might convey that you have the capacity which is in fact just the knowledge of how to do it.
Here are some examples:
Tú sabes conducir, pero no puedes hacerlo si no tienes licencia You know how to drive, but you can't/are not able or allowed to do it if you don't have a driving license
Si tienes permiso entonces puedes conducir If you have a license then you can drive
It implies not only that you have the knowledge to drive (you would not have a license if you didn't have the knowledge and passed your test), but that you have permission to do it (police won't fine you, because you are authorized to drive).
Sé cómo darte permiso de administrador en el sistema, pero no puedo hacerlo sin la autorización del jefe I know how to grant you administrator permissions in the system, but I can't do it without the boss' authorization
Yo puedo autorizarte a ser administrador del sistema, pero es Juan quien sabe qué hay que cambiar para hacerlo I can authorize you to be a system administrator, but it's Juan who knows what needs to be changed to do it
In this example the subject has capacity or authorization to do something, but not the knowledge.
Yo puedo trepar por esa cuerda I can climb that rope
I know how to do it and I imply and stress that I also have the physical strength to do it
Yo sé cómo funciona I know how it works
Si tienes un minuto puedo explicarte cómo funciona If you have a minute I can explain you how it works
I imply not only that I know how it works, but that I have the capacity or time to teach you that
Yo sé explicarte cómo funciona I know how to explain you how it works
poder in this sentence and the previous. The latter says that you know how to explain something, while the former says you have the capacity to explain it. That is the inflection of the difference between the two.
Yo sé cocinar ese plato, pero no puedo hacerlo sin los ingredientes I know how to cook that plate, but I can't do it without the ingredients
The tricky thing with this one is that probably no one would call on you if you had used
puedo for the first one ("puedo preparar ese plato"). You imply that you indeed know how to and can prepare the plate, but you wouldn't be able to do it without the ingredients.