Congratulations on beginning to understand what it means to be "in command". It does not mean that you will get everything right every time, but it does mean that you're learning what it is to make important decisions for yourself, which in this case you have done admirably.
Never mind all the rubbish about shutting down and feeling a prat, you'd feel an even bigger prat getting turned away from the Pearly Gates because you ignored the warning and continued with the go-around.
The only thing I've read in your post that gives me slight concern is the instructor's apparent willingness to send you off again in the same aeroplane without having first investigated the cause of the "cough".
"It sometimes does that on finals" strikes me as being rather belated and perhaps a little too casual.