What's the proper way to typeset a differential operator?

I can't seem to find any consensus on the right way to typeset a differential operator, whether it is:

• in a standalone context:
• as part of a derivative:
• as part of an integral:

In all of these cases, I have seen them sometimes italicized like variables (as in the second and third examples above) and sometimes not italicized like operator names (as in the first example above). There is also additional variation in how much spacing goes between the "d" and whatever it is acting on.

Seeing as this is an incredibly common symbol in many branches of mathematics, I am curious about the lack of standardization -- I've seen all possible combinations used in all possible contexts by many respected authors and publishers.

Are there any rules of thumb that I can follow? I personally tend to prefer the non-italicized version with very little space next to it, so I add:

\renewcommand{\d}[1]{\ensuremath{\operatorname{d}\!{#1}}}

to my default preamble, and use that everywhere, but I'd like to know if I'm breaking any hard-and-fast rules here.