I think this is the new default of the web site. Under the check it says : [Uncheck if on a shared computer]. You can modify the time before it log you out in the option of Yahoo mail. I guess it works on the other PC because there not considered has main PC. I found 2 yahoo related object on my pc.
C:\Documents and Settings\TCHP\Application Data\Macromedia\Flash Player\#SharedObjects\TW3XK8Z9\login.yahoo.com
C:\Documents and Settings\TCHP\Application Data\Macromedia\Flash Player\macromedia.com\support\flashplayer\sys\#login.yahoo.com
Both contain .sol file I'm not sure if they would really change something.
Asus Crosshair 1, AMD 4800+ x2, 4G RAM
BFG GTX 260, Acer AL2216W, ViewSonic VT2342
Haf Full tower, Watercool Thermatake / Cooler Master
3.3Tb ( 1gi Raid, 0.3 backup, 2tb External )
G13, G110, G500, X-540
750 W0117 Thermaltake Active Toughpower