Added later: remarks in the comments below lead me to believe that posting the the following excerpt from the introduction of [2] may help serve to alleviate any further confusion:

"It was observed long ago as 1932 (Dorroh's Theorem) that any non-unital ring $R$ may be embedded in a ring with unity. This is done by adjoining a copy of $\mathbb{Z}$, the ring of integers, to $R$. This does not preserve all the nice properties which $R$ might have, nor is it minimal in any of various senses; and so over the decades many embeddings have been invented to serve diverse purposes. For example, if $R$ is regular (or some generalization of regular such as $\pi$-regular) one would like to embed $R$ into a regular ring (or the generalization). There are other sorts of properties (semiprime, artinian domain, Ore domain) which one may wish to preserve in going from $R$ to some ring with 1, say $R^1$, all the while without adjoining anything more than necessary. It turns out that there is one construction [...] which will give all the main results as well as some new ones, although there is not yet one proof by which to do it. In the case of the generalized sorts of regularity, the ring formed [...] satisfies a universal property with respect to the adjunction of 1."

2 clarified relation to OP in light of comments

This is a slightly different kind of example, namely, one where multiple definitions are possible and there is no best choice generallythe original definition had to be revised when it later was realized that it was useless in certain contexts. Probably most readers have encountered the Dorroh extension [1] as a way to adjoin 1 to a rng (ring without unit). While various arguments can be made for the naturality of this construction, it turns out that this is the wrong definition in many contexts because it doesn't preserve crucial properties of the source rng and/or doesn't satisfy various minimality properties. For much further discussion see [2]. I mention it primarily as another perspective on the way that definitions may evolve. A word of warning: I've mentioned this many times over the years and almost always someone argues tooth-and-nail for the naturality of the Dorroh unital extension without first appreciating the issues that arise in contexts outside their expertise, e.g. see the AAA thread [3]. To avoid that here I highly recommend first perusing [2] before commenting.

[2] W.D. Burgess; P.N. Stewart.
The characteristic ring and the "best" way to adjoin a one.
J. Austral. Math. Soc. 47 (1989) 483-496 http://anziamj.austms.org.au/JAMSA/V47/Part3/Burgess/p0483.html

[3] rings, ideals and correspondence theorem -- clarification requested. Ask an Algebraist, 8/4/2008