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Sorry, no etymologies found.
If by "inspired by" Starbucks is suggesting the Latin "inspirare," to breathe or blow into, as in to breathe life into a new creation, perhaps.
Calling has to do with spirit, and both in biblical Hebrew and Greek, the word "spirit" can be rendered as "breath," or "breath of life," the breathing in and breathing out of that which inspires (from Latin, inspirare), and in human experience, that sort of transaction overflows the confines of what we've come to think of as religion.
Inspiration comes from the Latin, inspirare, with an original meaning to "breathe in spirit."
William of Tocco on this point are worth transcribing: "Erat enim novos in sua lectione movens articulos, novum modum et clarum determinandi inveniens, et novas reducens in determinationibus rationes: ut nemo qui ipsum audisset nova docere, et novis rationibus dubia definire dubitaret, quod eum Deus novi luminis radiis illustrasset, qui statim tam certi c [oe] pisset esse judicii, ut non dubitaret novas opiniones docere et scribere, quas Deus dignatus esset noviter inspirare."
Inspire - from the Latin: inspirare, "to fill with breath.
Manuale d’uso: inspirare, trattenere il fiato, espirare, godersi l’effetto.