Here's an audio recording of Hopper relating that story: https://archive.org/details/podcast_from-archives_grace-hopp... starting at 16:12.
You'll see she says "they found the failing relay" and "the operator got a pair of tweezers ... and below it he wrote “First actual case of bug being found.”"
https://chsi.harvard.edu/harvard-ibm-mark-1-language notes four terms which entered programming through the Mark I, not the Mark II. These are: "Loop", "Patch", "Library", and "Bug" - with images of the physical nature of the first three! For the latter it says (emphasis mine):
] The "Bug"
] Before 1944, electrical engineers already used the term "bug" to refer to hard-to-find physical defects that hindered the operation of an electric device. The Mark I team appropriated the term for unexpected problems in the "coding" of a problem. Above are cartoons drawn by Grace Hopper of the different types of bugs encountered during her work.
The paper "Stalking the Elusive Computer Bug" by Peggy Aldrich Kidwell (doi:10.1109/85.728224) date those cartoons to 1945, and gives other pre-Mark II examples of using 'bug' for the Mark I:
] I. B. Cohen has examined a photocopy of the logbook kept by Robert Campbell, a young physicist at the Computation Laboratory. It tersely summarizes hours spent finding and correcting errors. On 17 April 1944, Campbell wrote, "Ran test program. Mr. Durfee from I.B.M. was here to help us find 'bugs.'"
and Hopper writing "a debugging note to operators" in 1945/1946 and "errors in mathematics and tape bugs pursued and captured by Ensign Bloch and Ensign Campbell."