Proper punctuation after cases environment

You've pretty much enumerated all the sensible options in your question, so I don't know that there's anything else. What I've seen sometimes (IIRC) is to put a comma after each case, and then I guess the period could go at the end of the last one.

This is what I would do:

Absolute value is defined as:
%
\[
\left\lvert x \right\rvert \coloneqq
 \begin{cases}
  x & \text{if } x \ge 0 \\
 -x & \text{if } x < 0
 \end{cases}
\]

Thus: a colon introducing the maths, and no punctuation afterwards. My reasoning is in my answer to ... err ... I thought we had a question on punctuation after maths here? What happened to it? Nonetheless, it was asked over on MathOverflow so I can cite it as this answer instead.

My basic point is that anyone reading more than a few symbols will need to stop "reading English" and switch to "reading maths" and that this mental switch has the same effect as a full stop. Of course, you can punctuate internally to the maths since punctuation still has meaning, but it should be read as being internal to the mathematics and not refer to the external text. Thus David's suggestion in his last sentence (at time of writing) makes sense to me.

Those that find the absence of a full stop abhorrent (there are some, as can be seen by looking at the comments to my answer on MO) can adjust the preceding remark to make it a full sentence, or to make it clearer that it isn't. Thus:

Absolute value is defined as in the following formula.
%
\[
\left\lvert x \right\rvert \coloneqq
 \begin{cases}
  x & \text{if } x \ge 0 \\
 -x & \text{if } x < 0
\end{cases}
\]

(You may notice that I've also introduced a couple of other stylistic "improvements"!)