Blood urea nitrogen and usage of proteins?
The quotes in the question provide incomplete information at best.

Let's focus on the nitrogen (N) that comes from protein, ignoring for now how metabolism differs among amino acids. We'll assume for simplicity that liver function is OK so that the urea N production rate is a reasonable measure of protein N degradation rate (say in units like moles of N per hour).

First, consider simple conservation of mass:


  Intake = Loss + Net Storage


If you think about this in terms of protein N from the perspective of the whole body, you start to see the limitations in those quotes. 

First, loss from the whole body of metabolized protein N in the form of urea N mostly occurs via the urine. Impaired kidney function may lead to less urea being excreted than is being formed. You will then have "Net Storage" of urea N in the blood and body fluids, so BUN can rise even if metabolism of protein is unchanged. In fact, BUN tests can be used with other tests to provide information about kidney function. Thus BUN levels don't necessarily tell you much about the urea N production rate or protein degradation rate.

Second, you have to consider that one "use" of proteins can be "Net Storage" to build muscle mass. That type of "use" won't show up in increased urea N production. Alternatively, body protein in muscle can be broken down to be "used" for metabolism/catabolism during an overnight fast, possibly leading to increased urea N production rate depending on how the mix of protein, fat, and carbohydrate catabolism has changed. Furthermore, the effects on BUN levels will depend on kidney function. 

Both storage and degradation/catabolism are important "uses" of protein N, but they have different implications for production of urea N, and BUN levels don't necessarily represent the urea production rate.