I'm wondering what could be the motivation behind the term "trace", especially in the simple case of a trace of a matrix.
Looking through some geometric interpretations and bearing in mind the traditional meanings of the word "trace", I could not conceive a satisfactory answer.
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The origin of the (German) terminology Spur, which apparently during the 20h century was translated to English as "trace", is now laid out in the answer to this question on hsm.stackexchange.
Basically, Dedekind introduced what in modern terminology is the field trace as "Spur" on page 5 of his article Über die Discriminanten endlicher Körper. (In: Abhandlungen der Königlichen Gesellschaft der Wissenschaften in Göttingen, 1882), as a third rational invariant for algebraic numbers, after the already known discriminants and norms in algebraic field extensions.
Frobenius extended the use ("Following Dedekind") in his 1899 report Über die Composition der Charaktere einer Gruppe, where he takes the sum-of-diagonal definition for any matrix, not just ones coming from field elements, and might have been the first to do so.
So for original motivation, you would have to ask Dedekind, which is impossible as of 2022.
If I may speculate, for Dedekind the sum of diagonal elements was more a computational afterthought. He was defining, after already knowing norms and discriminants, a new rational invariant of an algebraic number, which sometimes vanishes (e.g. for $\sqrt{-2}$ or $\sqrt[5]{3}$) and sometimes does not (e.g. for $4+\sqrt[3]{2}$). To call such an invariant the element's "trace" (which it leaves in the ground field, for us to see and get a first idea of the element) seems not totally unmotivated.
Trace is a translation from the German "die Spur" and Google translate gives as "the track" which is similar to the definition of trace given by "follow or mark the course or position of".