No, It said [DOWNLOAD] on Appstore, so it downloaded the installer before running the installation, so it wasn't on my computer. Why it downloaded the old version, is beyond me. It should have downloaded the latest released beta.
Anyhow, I went and downloaded the Public beta, instead, and this time I did the whole thing of redeeming the code and downloading the installer. This time (I thought) is going to be right, it will download and install the latest.
Well, to my amazement, after installation completed, it was not the latest. Then the update to PB6 appeared in App Store, but this time it downloaded it. Thanks God. I was so worried.
This installation of Beta software via App Store is not very stable, I believe. It seems to be better for the Public betas, but for developers, I believe we better stay with the downloaded package.