The 3-button controller and the 6-button controller are entirely different things, and unfortunately the 6-button controller does not have a 3-button mode.
Specifically, the 3-button controller's III button is a programmable button which can act like either the Run or Select buttons. So, it's useful for any game that puts an important gameplay function on Run or Select, which some games did. It has a switch on it to switch between Run or Select. It's most useful for Frgotten Worlds and Barunba (HuCard shmup which also has rotating fire), but there are other games that benefit from 3-button controllers as well.
The 6-button controller, though, is a proper 6-button controller, with a 2-button mode for compatibility, and 6-button mode for games that support it. Unfortunately the PCE doesn't have anywhere near as many 6-button pad compatible games as the Genesis/Sega CD/32X do, though. While the Genesis, Sega CD, and 32X have between them several dozen games that have 6-button support, the support list for the PCE 6-button pad is much shorter:
http://www.pcenginefx.com/forums/index.php?topic=764 It's some fighting games -- Strip Fighter II and Street Fighter II for HuCard, Advanced V.G. and Martial Champion for Super CD, and all eight fighting and wrestling games for Arcade CD -- plus some RPGs and mahjong games that support the 6-button controller for menu shortcuts and the like, and that's it.