The etymology of the word tracheotomy comes from two Greek words: the root tom- (from Greek τομή tomḗ) meaning "to cut", and the word trachea (from Greek τραχεία tracheía). The word tracheostomy, including the root stom- (from Greek στόμα stóma) meaning "mouth," refers to the making of a semi-permanent or permanent opening, and to the opening itself. Some sources offer different definitions of the above terms. Part of the ambiguity is due to the uncertainty of the intended permanence of the stoma (hole) at the time it is created.
(I have done tracheotomys by the book. I leave tracheostomys to physicians. )