What does the outbound path processor do?
Inbound path processing is used for incoming requests to the site. Outbound path processing is used when generating URL's, doesn't matter if it is outside of the site or not, in fact it would be very uncommon to use it for those kind of links.

The most obvious example is language negotation path prefix/domain, on an incoming request to en/node/1, inbound path processing removes "en" and sets the current language to english, when generating a linke to "node/1", path outbound processing adds the en/ prefix based on the current or provided language.