The GRS is always drifting in longitude, due to the East-West wind movement in the belts and West-East wind movement in the zones.
The software "shows" the position of the GRS based on its current longitude when it crosses the "middle" (Central Meridian) of the planet. As explained above, it's always changing and so the software needs to be regularly (once every few months) updated with the "current" longitude of the GRS so that it shows it transiting the CM at the correct time.
The current value of the longitude of the GRS is 119° in System II. So see if your software has the ability to either automatically or manually update this value - then you'll ge the GRS showing at the correct position at the correct time.
That S&T page is accurate and can be used to predict transit times of the GRS, otherwise you could download a free program called
Jupiter 2 which is what I use. It lets you update the position of the GRS, and gives you visual (and ephemeris) representation of the GRS as well as the position of the galilean moons so you can plan in advance (or check in retrospect) transits and shadow transits. I use it almost daily when the skies are clear.
Hope that helps.