Judging by Lewis and Short (and other sources), planeta means a wandering star and was borrowed from Greek. Apparently the word is post-classical, and the classical expression is stellae errantes/erraticae/errones, used (almost?) exclusively in plural.
When the word planeta enters Latin literature, is it defined? Is it ever explicitly linked to the classical version mentioned in L&S? I would like to know what the word meant in Latin. (I prefer classical Latin, but if the word is younger than that, I won't require it.) If the meaning evolved over time, an account of this evolution would be great.
My best guess is that it means any bright spots in the sky that move. That may or may not include the Sun and the Moon, but it most certainly does not include other actual stars than our own. I would assume the definition should be extended to objects visible with a telescope but too dim for the naked eye, although such conclusions can be hard to extrapolate from extant literature. Perhaps comets are planetae too. One of the most famous test cases is Pluto, whose planetship has been revoked. Would Pluto be a planeta using the early meaning of the word?
I think that in modern Latin it is wisest to use the word planeta to mean precisely the same thing as "planet" to avoid confusions. This definition is almost certainly different from the earlier meaning.