Is there a simple solution to force my Windows XP machine to open a program on this monitor where I see the desktop shortcut?
The UltraMon utility has this functionality.
The shortcuts can be accessed from the UltraMon menu, or as standard desktop shortcuts.
One trick is to uncheck "extend my display" to your second monitor, open the program, and then recheck "extend my display". It will force it onto the primary monitor. You might need to try restarting/disconnecting in between the checks. Unfortunately, if you want it on the other screen, bwall's solution may be the best. XP is kinda bad for dual monitors. Windows 7 has a pretty decent success with 'remembering' where you put the windows last, but for XP it decides at the tip of a hat it seems. =(