> The mirror process seems to take about 1-2 seconds,
Probably didn't get anything, see the log file.

> C:/My Web Sites/www.EXAMPLE.com/www.EXAMPLE.com/ index.html
everything is stored in C:/My Web Sites/www.EXAMPLE.com and below.

> files are on my PC, I think I'm still browsing the web
Probably are.  Set options -> Build -> No external pages =  checked and you'll
get a warning page if you click on a non-mirrored link instead of going to the
net.

You can also look at the url displayed in your browser.
net files urls look like <http://www.EXAMPLE.com/>... while local files are
<file://C:/my%20Web%20Sites/www.EXAMPLE.com/>...