Consider that the word star is indeed used in British English and American English for anything that you can, let's say, "see in the sky" (even if it has literally nothing to do with A Sun).
Great examples are words like "Telstar" and "OnStar" ...
So, it's commonplace to have sentences such as "What's that star - actually that's the space shuttle" or indeed just "that star there is Jupiter". Indeed, consider, before modern times every "point of light" in the sky was a star - whether actually a galaxy, planet, "shooting star", actually a Sun, or anything else.
(The term "shooting star" particularly gives the feel of the more archaic, poetic, meaning of "star".)
I appreciate that the death star is a large object (more like a moon) when seen from relatively close - but then, a Sun, or indeed a GPS satellite, is "large" when seen from a certain point. You can easily imagine the Death Star being a twinkling point from another planet in it's construction system - So, picture this, we're on Earth and it's a slowly-growing "new moon of Jupiter" - quite ominous; a death star indeed.
Consider too that simply, in the very title "Star Wars" the word "Star" doesn't literally mean the hot burning balls of gas - if you think it through, it more means "planetary systems": it's literally the planets (the dwellers thereon) warring, it's not literally the Suns or anyone living on the surface of the Suns, warring!
"Planet Wars" or perhaps "planetary-system wars!" would "make more sense" but doesn't sound as good!
Thus, an excellent answer to your question is this:
Canonically, "star" in "Death Star" is being used in exactly same poetic, somewhat archaic way, as: in the very title of the series.
And that's pretty canon, right?! :)