People suffering from Anorexia also purge (i.e. self-induced vomiting), so no she is not bulimic. As a matter of fact, Anorexia is usually divided into two subtypes, the restrictive type (those who refuse to eat or heavily restrict their caloric intake) and the purging type (those who use laxatives, self-induce vomiting, exercise excessively to compensate, etc). Even if truthfully, people with Anorexia most often fluctuate between both subtypes during the course of the condition or even have both at the same time.
For her to have been bulimic, she would have had to also have binge eating phases (like it was implied for the mother). That's the difference between both conditions. Bulimia is an eating disorder characterized by binge eating phases followed by purging. Anorexia on the contrary, no matter which subtype, does not indulge in binge eating.
To be fair however, most people with EDs often go through different types of EDs phases during their lives and be bulimic at one point in their lives, anorexic some time before or after, or even become a binge eater at another point in their lives (or even all at the same time). Afterall, the different types of EDs are most often just different ways to express the same discomfort someone feels about their lives.
So from what we have seen so far - self-induced vomiting, restrictive behaviours (making excuses not to eat, not eating much, etc), no binge eating phases - I think we can safely say that this is indeed Anorexia and not Bulimia.
Her condition is BULIMIA not ANOREXIA