I will try to answer your specific question in a general manner using my legal background in Family Law:

what legally constitutes a "separation"

**the action or state of moving or being moved apart but still connected.**

*Separation is not same as divorce. So the husband is still considered to be married but separated. The court will approve the separation agreement allowing the couples to live apart and slowly try to resolve conflict and fix the marriage*

**Divorce ends the marriage completely and will allow both parties to enter into 
 intimate relationship and remarry, whereas Separation does not allow this.**

 Can one spouse just run away and declare this a "separation"?

**NO. In order for a separation to be declared the court first must approve the separation.** 

*Before you can legally live apart you have to enter into a court declared separation agreement, not the other way around.*

What the man did in the description is not separation, it is **abandonment**.

As a result, she has fallen very sick because of all of the pressure.

*Family Law matters are always stressful when you don't know the court-process. Court litigation makes everyone nervous and stress. So try to calm them and ease them with comfort. They should retain an attorney or continue to do research on your own to learn about the process.*