(Looks as some of what I have to say here was answered by others before I posted -- some good info above, particularly Mr. Lerch.)
I think the unfortunate quick answer is "They won't." Similarly to the phenomenon of users owning product manuals but refusing to open them, a sudden, surprising, out-of-context (well, to the base user at least) message is going to be sudden, surprising, and out-of-context.
My inherit pessimism aside, I've found anecdotal success with the spirit of your given suggestions. WRT the first point, I agree with "short" but not as much "precise." The non-technical user is going to have issues with any error message. A "precise" message even more so, because the non-technical user needs not less information, but more (i.e., needs a bit of education) to actually understand what happened.
As an example, in the second bullet point, the non-technical user is going to have issues with the first part of your sample message.
- What is it connecting to?
- What server?
- Why is it unable?
- Did I do something wrong?
- Did the software do something wrong?
Some or all of these questions are going to run through their head before they really get around to comprehending the suggested response. When they regain their composure and read the second part, they'll have another bunch of questions about how to check their internet connection. Some will interpret it as a modem or router issue and power-cycle that whole stack (no joke).
A bunch of this isn't really solvable in the error message; people have a level of computer literacy and your application probably won't have a signifcant impact on that. But if you were to
- Provide first as simple a sentence you can explaining why they're seeing the message. 5-7 words.
- Use simple, conversational language to explain in a bit more detail.
- Offer (if it's the site's fault) an appology
- Find a clever way to tell them not to panic
- Ensure them things will get better
you may make the experience a bit more pleasant.
Note, the last few are easy to get wrong and end up sounding patronizing, so be cautious and test against some friends first.