IMHO, this is exactly what the issue is. When my market app upgraded to play, there was no indication that the actual name of the app was changing also. I'm running MIUI with the ADW Launcher and the upgrade also didn't remove the market app icon on the desktop.
When I tapped the Market icon to launch the market, nothing happened and after the first tap, the icon disappeared. Thinking WTF? I checked all apps and it was nowhere to be found. Having had this issue before with different launchers where an icon will exist on one launcher but not another, I switched to Go and the Market icon wasn't there either.
I hopped online to download an apk for Market when I stumbled onto something that indicated it might be "Play" now instead. Sure enough, the icon was there all the time, it's just now called "Play".
Not a very user friendly upgrade/name change of the Market app if you ask me.