I think this is an excellent distinction. Some associations:
Yesterday in the gym I observed in practice for the first time - or maybe re-observed after forgetting - the difference between pain and suffering. I was doing a relatively difficult weight session (for me) and observed how I was in physical pain, and even a bit of mental strain, but was not suffering. I wanted to do it, I saw the value of doing it, so I even felt happy and proud, as the article suggests.
I am reminded of this diagram from anxiety-curing circles.
The "comfort zone" is the no-stress and no-growth zone, maybe even the backtracking zone. The "learning zone" is the manageable-stress and growth zone, where we want to be, what I believe the article would call "eustress". The "panic zone" is the overwhelm zone, where the article might say that we are overwhelmed and lack the resources to cope. With practice and experience, the circles expand.
I sometimes sleepover at my mother's. As maybe many mothers do, she tends to want to do everything for me, keep me happy and safe. I observe that even after just a couple of days, I seem to lose a bit of strength and resilience.
In conclusion, I agree with you and the article, it is good to be challenged and to keep growing, and I believe that the best safety is having the internal resources to face challenges. However, to my mind, this does not mean that we should adopt a hyper-individualist attitude, since overwhelm, breakdown, and panic do not lead to anything good. Apparently, as usual, the middle-way applies here too 
My experiences from my working life were that there was a certain amount of “positive stress” which could spur one on to do better work. But if you go over the line and life piles a lot of stressful circumstances on you, it can become sharply negative, your body and mind may rebel against things.
Jeroen
I was introduced to a new concept today that helped clarify things for me. Positive (eustress) vs negative (distress) stress.
https://www.medicalnewstoday.com/articles/eustress-vs-distress#summary
For a while I've been embracing the value of challenge and difficulty in their ability, and I'd say necessity, to improve mental and physical health. The human mind and body are built to react positively to certain levels of stress. Like the immune or skeletal system, live too long in a sterile or gravityless environment and these systems degenerate and break down. I think there is truth to this regarding our mental well being too. Remove all challenges from someone's life and they become incapable of coping with the inevitabilities of life, or something like how really beautiful people don't have to try as hard for attention and they develop comparatively boring personalities.
Many people face real struggles in life that wear them down and degrade their quality of life. Being able to make this distinction between types of stress helps.
The main distinction between the two has to do with the level of resources, both external and internal, the individual facing the stress has to meet it. From a Buddhist perspective in developing a compassion practice wishing for people to have the internal resources, such as resilience and confidence to face life's challenges, has been more conducive to building compassion for me than things like wishing for things like safety. We all can do things in our lives to increase our ability to meet difficulty, changing distress into eustress. When I get overwhelmed by situations I consciously turn my thinking from worry into focusing on the things I can do to improve the situation and my mental health improves.
person
In my experience there's a real interplay between thoughts and feelings. The feeling in the body gives rise to certain thought patterns and those thought patterns reinforce the feeling state. By letting go of thoughts at times, seeing them for what they are and dropping into the bodily feeling, those feelings begin to release and wind down.
And not identifying with thoughts and having compassion for the difficulty of thoughts that do arise are both really excellent practices in my experience.
I think a better word than false for thoughts is illusory. A mirage is illusory, the details of the appearance can be more or less correct but on closer inspection it all dissolves away. Like 2+2=4 and 2+2=5 are both thoughts, one is more true than the other but they are both illusory.
person