*Disclaimer* I am not a psychologist but I have studied psychology as a supplementary study. Further validation would be necessary.
From what I know and understand, there are 2 kinds of stress, eustress (good) and distress (bad). Our bodies cannot exactly differentiate between the two (they come from the same place in the brain). Stress could be defined as the relationship and reaction to an abnormal trigger (cannot remember the specific term right now). The reaction itself is the "catalyst" to solve the problem, normally defined by psychological problems such as acute, episodic, and chronic stress or relative to the length of time used to go back to neutral stress or equilibrium. Technically, all health problems can be traced to stress, even physical ailments.
To be diagnosed with anxiety may be relative to the length of time and what causes the stress. Most people experience stress so in a way, anxiety is both already the stress and caused by stress. For example, stage fright may cause anxiety but an individual may be stressed by large crowds constantly which may be more chronic or it may be only in relation to a "spotlight" environment and thus the anxiety may be more episodic. Anxiety is also used as a term to describe another psychological issue such as a phobia. Using the crowd example, the amount of people or the spotlight may not be the problem but claustrophobia may be described with anxiety as a feeling of stress.
I do not normally hear people use the term "they are in eustress." Distress is more commonly used but, at least in the English language, can be equivalent to flustered which is not necessarily the same thing as more problematic forms of stress like anxiety. In fact, anyone can be diagnosed with any varying form of a psychological disorder. What makes it "official" is when the individual realizes/notices that it is affecting their life. For example, in relation to stress, because all people feel stress, they may not feel inclined to go to a psychologist until it is problematic, as in they can no longer function in their day to day life because it gets in the way of it, i.e. work, family, friends, food, etc.
In short, in answer to your question, Anxiety and stress may be defined more on the application it is being used. Anxiety is much different than Chronic Stress (as a diagnosis) but the words are often used as a way to describe the kind or amount of stress one is having, especially in any kind of distress.
The Yerkes Dodson law https://en.wikipedia.org/wiki/Yerkes%E2%80%93Dodson_law shows that increased arousal over periods of time impair performance and is experienced as stress. Best performance is when there is a challenge, but it is engaging and managable. If however there are too many challenges the person becomes overwhelmed and this is what we call stress. What is 'too much' will be experienced differently at times, depending on the individual's resilience, capabilities (which can be impaired by illness or whatever), and normal coping styles.
Anxiety can contribute to stress levels, as anxiety is in effect a sense of threat, and so leads to physiological arousal in itself. Anxiety is about a threat which will happen, something that is percieved as being in the future - it may be sooner or later.
Lovibond and Lovibond's 1995 Depression Anxiety and Stress Scale differentiates nicely between anxiety and stress. http://www.womenshealthapta.org/wp-content/uploads/2013/12/DASS-Lovibond.pdf