As others have noted, each carrier/program has its own interpretation for award tickets. Generally, the domestic stopover permitted on an overseas award ticket is at the NAmerican gateway city, that is the airport at which you connect to your overseas flight. Some carriers/programs may, however, permit a mid-point stopover as long as it is on a permitted routing. In many cases, this is rather difficult when most carriers still operate hub and spoke route systems.

For example, if you live in BUF and are flying UA to SIN on an award, your routing would be via one of their major overseas hub airports: JFK, ORD, SEA, LAX or SFO. You'd be permitted a stopover at any of these, and you might de able to do a stop-over in DEN if you flew BUF-ORD-DEN-SEA/SFO/LAX-NRT/HKG-SIN. But you most likely could not do a stop-over if you wanted to so it in Kansas City, which is considerably off-line [BUF-ORD-KCI-ORD-NRT/HKG-SIN].

At least that's how I would interpret the matter based on my experience with awards on several different carriers/programs.