Final answer:
Hildegard of Bingen is the correct answer as she was an abbess, composer, and a visionary active in religious and diplomatic affairs, not to be confused with the other women composers mentioned who lived in different times and had different life circumstances.
Step-by-step explanation:
The first woman composer known to have left a significant number of works that have survived, and who also served as abbess of the convent at Rupertsberg, was a visionary and mystic active in various spheres including religious and diplomatic affairs. She is none other than Hildegard of Bingen. Her multifaceted life included being a composer, healer, and theologian, and she was influential in many aspects of religious and monastic life of the time.
Hildegard’s contributions to music and the breadth of her work were significant, especially considering the era in which she lived. She composed music and wrote texts for liturgical songs, which were an integral part of her large-scale musical drama, Ordo Virtutum. Her compositions remain important in the historical context of sacred music and are evidence of the unique role that women in monastic communities could play in the arts and wider societal affairs during the Medieval period.
While other options mentioned, such as Clara Schumann, Fanny Mendelssohn, and Barbara Strozzi, were also important women composers, they lived in different periods and do not fit the description of being the abbess of a convent and a medieval visionary as Hildegard of Bingen does.