In others (such as this one), the score is shifted by a half-note, so it starts with an upbeat. Note that this induces syncopated harmony. (If you have the same harmony on a weak beat and on the following strong beat, it is called syncopated harmony, as in 'D' on this example). This is sometimes considered an error in music theory. Is this a wrong edition?
Actually the score is somewhat ambiguous, since the first repeat marks end with a half rest, which is not necessary with the upbeat. (One would likely make a short rest, no matter which notation is used). Given the slight emphasis on beats 1 and 3 is unchanged under this modification, the difference is not big, if noticeable at all.
It is often discussed, to which degree old scores should be revised to accomodate the expectations and habits of today's readers. This might be a weakly justified one.
Palestrina slightly predated the widespread adoption of of modern measure lines ("bars") in written music, which didn't take place until the 17th century, so this piece was probably first transcribed without measure lines.
With that being the case, it's largely a matter of opinion whether you draw the lines so this song begins on the half-note upbeat or the downbeat. Arguably, rendering it with any measure lines is "wrong" in terms of strict authenticity (although you could also make a strong case that earlier composers thought in measures, they just didn't write in them).
As far as the harmony --as well as the whole note held across a bar line, it may indicate that someone less well versed in music theory drew the measure lines for the second version. However, music theory has changed greatly over the centuries. If those conventions were NOT typical for Palestrina, it's also plausible that later transcribers may have been "wrongly" influenced by them to start the piece on an upbeat.