Well-definedness of uncomputable functions. I have been reading about Rayo's function and uncomputable functions in general, and have gotten very confused. There is apparently concern over the well-definedness of Rayo's function, but I never had any doubt that things such as the busy beaver function were well-defined. Then I read this article where it was stated that two numbers described by uncomputable functions like BB might be incomparable within a certain system of axioms - it could be independent of the system which one was larger. This was taken to mean that one might not be larger than the other at all. I took this to suggest that even a simple uncomputable function like BB(n) could be undefined in a sense.

However, it seems to me that, given two numbers, it should always be possible to decide which is bigger. It also seems like the busy beaver function should be well-defined since it is making a precise statement about machines which could theoretically be implemented in the physical world.

So, are uncomputable functions well-defined in general? What characteristics would make one well-defined or not, and is there something I'm conceptually missing in the way that I am approaching these questions? Thank you for helping me understand!
The busy-beaver-function is well-defined because every halting turing machine can only write a finite number of ones on the tape, hence there must be a champion, and $bb(n)$ is the number of ones this champion writes down.

But the busy-beaver-function is not computable because we cannot determine which turing machines halt and which do not. This (and only this) prevents us from determining the busy-beaver-champion and therefore determining $bb(n)$ in general.

"Computable" means that there is an algorithm being able for every n to either determine $f(n)$ or determine that $f(n)$ is not defined.

"Well-defined" means that there is a description guaranteeing that for every $n$ there is either only one possible value $f(n)$ or that the function is not defined.  (It is not necessary that this $f(n)$ can actually be determined , if $f(n)$ is defined.)
The busy-beaver-function is well-defined because every halting turing machine can only write a finite number of ones on the tape, hence there must be a champion, and $bb(n)$ is the number of ones this champion writes down.

But the busy-beaver-function is not computable because we cannot determine which turing machines halt and which do not. This (and only this) prevents us from determining the busy-beaver-champion and therefore determining $bb(n)$ in general.

"Computable" means that there is an algorithm being able for every n to either determine $f(n)$ or determine that $f(n)$ is not defined.

"Well-defined" means that there is a description guaranteeing that for every $n$ there is either only one possible value $f(n)$ or that the function is not defined.  (It is not necessary that this $f(n)$ can actually be determined , if $f(n)$ is defined.)