In a way, it's only a matter of preference. A good "canonical" form must be such that there is exactly one such form for any given conic. The form you mention has that property, and it is easy to read off the conic's parameters from it.

The two equations you give are of the same form. They only represent the same circle for as long as and are related by the translation you speak of.

It's really a matter of preference. I could also choose to write a conic as , and this form is also unique.