I found a solution. And, yes, it was a Display setting rather than a WMP setting.
quote, from
http://www.peterprovost.org/archive/2003/08/11/693.aspx :
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This is a feature of your video card. nVidia, if I'm guessing right (it's the most popular brand that does it). Instructions, presuming an nVidia card and moderately recent Detonator drivers: Bring up Desktop properties, on the "Settings" tab, click "Advanced". On the advanced settings dialog, click the nVidia tab (for instance, mine has a little nVidia eye log and the text "GeForce Ti 4400"). That spawns a little "side" selector window. In that, click on "Overlay Controls". There is a section titled "Full Screen Video Mirroring Controls". Inside the "Full Screen Device" drop down, you should be able to pick which monitor, or "Disable" to turn the feature off.
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