Are you sure you had both 32 & 64 bit WMP on the same computer? I'm not sure if that is possible but I'll take your word for it. Check in "C:\Program Files\Windows Media Player" for the 64-bit version and "C:\Program Files (x86)\Windows Media Player" for the 32-bit version. Look thru the folder to find wmplayer.exe, right-click and select "Pin to Start Menu".
Both 32-bit and 64-bit versions should be able to play the same music files. Most likely one version has some codecs installed that the other doesn't.
In order to avoid chasing down codecs for different media formats, many people use VLC as their media player which includes codecs for pretty much any media format.