So in that case, BPR is only related to the hotstream and any coldstream airflow that is created by the first fan or compressor stage and not the coldstream airflow that the engine might produce outside the engine cowling?
No, this is not true.
The BPR is the ratio of the cold stream flow (which passes through the by-pass duct), to the hot stream flow (which passes through the engine core).
It is also worth noting that only a small fraction of the hot stream (core flow) is actually used in combustion, but it all passes through the turbines.
Strictly speaking the term BPR is not applicable to engine/propeller combinations, and this appears to be where you are going wrong.
But that does not mean that the matter should not be discussed in ATPL classes. It should be discussed, because it is fundamental to the subjects of stall/surge avoidance and also propulsive efficiency.
To teach the subject you should be fully familiar with how the concept of by-pass developed. It was first used to reduce the problems of compressor surge/stall, then was gradually developed to increase propulsive efficiency.
In that case, how do you explain the Ultra High Bypass engine? That is, where do you draw the line between an unducted fan and a propeller?
That is a very valid question.
The engineers that designed the first ultra-low by-pass engines (they were known in some circles as "leaky trubojets" ) to reduce surge/stall couldn’t possibly have envisaged the development of high by-pass turbofans or unducted fans. So we should not be surprised if we are now coming up with problems in deciding how to classify such engines.
EDIT: I know I'm splitting hairs a bit here, but that is what my students tend to do, so I'd really like to be able to give them a specific and detailed answer.
It sounds like you are experiencing the truth in the oldsaying that “The best way to learn something is to teach it.”
In order to be able to teach a subject properly your level of knowledge and understanding of the subject really needs to be at least one level above that at which you are teaching. This provides a bit of a buffer to help deal with the more gifted (probing) students.
A good place to start reading would be the book “The Jet Engine” by Rolls Royce. It covers the subject at the appropriate level for ATPL studies, without burying you in too much mathematics.