Update Market Application to Google Play Market Application I have the problem with latest update which re-brands "Market" app 
to "Google Play Market" app.

At one moment "Market" icon disappeared from both home screen and applications. 
I was seeking for the solution to that, found some (weird at the time) "Play Market" update installed 
and have uninstalled it. This brought back "Market" application.

Later I have seen that this was a silent patch which I should not uninstall!

Problem is, that, when I start "Market" app it ONLY displays following page in browser:
https://play.google.com/about/android-developer-policies.html

I am stuck here with this page, and the app is not updating again to "Play Market"

Does anyone know how to resolve this?
Try clearing the data for the Market and rebooting the phone.  Then log in to the Market, wait a minute, then back out and continue using your phone with the internet connection active.  Hopefully the Market should update itself within a few minutes.

Or, you can just download and install the .apk.  I rely on Android Police for the latest version; they've posted 3.9.16 here.