I think the point they are making here is that clearly stating what went wrong doesn't necessitate using "technical jargon".
Now, "your credentials have been denied" seems pretty clear and does not use jargon in my opinion, but telling the user "the ajax request failed, returning a 403 http error code" seems unhelpful and doesn't tell them what happened.
Even in your example there's a world of difference. "Your credentials have been denied" implies a problem with credentials, while 403 clearly states that the credentials are valid, they are just denied access to this resource.
I know it is a made up example, but it does show the problem with "dumbing down" the error messages. Details matter.
...then they clearly didn't make their point at all.
Big error (in communication) on their part.
Your single (2nd) sentence communicates everything required.
Now, "your credentials have been denied" seems pretty clear and does not use jargon in my opinion, but telling the user "the ajax request failed, returning a 403 http error code" seems unhelpful and doesn't tell them what happened.