[wesselbindt]

The first part of your comment is completely correct. The latter is a matter of taste, of course. I think the main thing that can be said for a lot of the definitions we have in algebra is that the ones we're using are the ones that stood the test of time because they turned out to be useful. The distinction between invertible elements (units) and irreducible elements, while complicated, also gave us a conceptual framework allowing us to prove lots of interesting and useful theorems.