All of these responses, as you noted in your question, focus on solving the problems that lead to your uncomfortable encounters, in an effort to reduce or eliminate the encounters with your bosses. While you may need help with your approach on that, I think a more useful response is how you deal with the encounter itself - because in the world of engineering, you are likely to always face things like this. Bosses that don't exactly understand what you do, situations that arise without warning and meetings where you are supposed to talk about it.
The key here is to become the expert that they are asking you to be. Highlight your clear concern for the problem, and reassure your bosses that you are the guy for the job. Show them the experience you are gaining, the dedication you have and the effort you are putting in to make progress. They may not want to hire a new person knowing that they are investing a lot in you, in terms of time and experience - and your job is to convince them that they can't find anyone better.
First, if during the meetings you review your technical approach to the problem, steps you've taken, results, etc. and that does not end the meeting, then you were too technical. Your audience did not understand how what you said relates to their problem, and for all they know you are making it up. That does not build trust. You need to find a practice audience to explain the situation to that is equally technical to your bosses, and hopefully with other similarities. If that person can understand it, then maybe your bosses can.
Second, I don't know if you are doing this, but it sounds like you also need regular communication with them on these issues. It would probably be helpful to have weekly/bi-weekly/monthly meetings on your progress, and you lead the meeting. This will help you practice your communication, it will demonstrate to them that you care about the issues, it will demonstrate your sincerity in resolving the issues and it will also allow them time to think about and understand the problems when they are not in "crisis mode" and probably cannot focus on your work as much as they are on the crisis. Also, it will put you in the role of "authority" on the matter - in the other meetings, they lead and you get thrown around. With this, you are the lead.
If they refuse to do this or "don't have time" and cancel, make the request anyway, send them written reports anyway - on time, every time. Even if they throw them away, at least they have that as a reminder that you are putting in an effort. This may help with their technical education - you can force them to be more technical, but maybe they will pick up on a few things with these meetings. And even if they cancel, you were the one with the request - you were in front of the problem.
Third, start the crisis conversation with, "I've given you updates up to date X, and here's what I've learned since then." And add, "And I still can't guarantee that this won't happen again. Maybe this experience will lead to an answer, however, here is the downward trend to the occurrences, etc..." You mention that the issues are less frequent, saving money, etc. Quality control is about reducing error, not eliminating it. If you know that you have reduced the problem, then you need a chart or graph or something to demonstrate that. These meetings can be a reminder that you have improved the situation and, in that regard, you are the right guy for the job.
Fourth, you need to have several "to do" items when you walk out of the crisis meeting. Tell your bosses you will follow-up with an assessment / report of the current situation by tomorrow / next week. Be sure to send that report or have that meeting. Schedule it before they leave if you can - and if they cancel, send a report anyway. Show them that you are worried about this even when they are not and even after the crisis is over. Fundamentally, that is what they pay you to do. Worry about it and hopefully solve it.
Similarly, make it a point to do another follow-up later - maybe a week or month later. Sincerely analyze the problem, the data you got and then communicate the direct actions you took and/or things you did different based on the new experience.
At some point, they will be saturated with data, meetings and information about the issue. They will start to understand you, what you are doing and why they need you. Or else you will gain some great skills and be able to take those to a place where your work is better appreciated.
The key here is to learn how to handle the situation. Your bosses know that you are stressed by the meetings. They are being bullies. Bullies love it when you are scared because that gives them the control. When you start asking for more meetings, you hand them reports and you become confident that no one else can do this job better than you, then you begin to be in control and will start to have better meetings (or else you need to leave anyway).