Game won't update: difference between Play Store and manual Install ? (Plants vs Zombies 2) my "Plants vs Zombies 2" won't update (it never did).

I wonder if the reason is that, for the rush to have the game, I probably installed it manually downloading it from the internet.

BUT, Play Store says the game is already installed, and won't update it.

Do I have to manually remove it and then install it ?
How do I have to backup my savegame ?
My guess is that you have to remove it, then reinstall it from the Playstore. It will probably see the installed file on the phone, but your Playstore account doesn't know about this, and won't activate the update. 

You can use Titanium Backup to backup the game, if your phone is rooted. Titanium Backup can backup apps and their data, and restore either of them separately. If the phone is not rooted, I don't know how to do this if the app doesn't have an export function. 

But... probably it won't overwrite your personal data. I can't give you any guarantees, but you probably won't lose your personal settings.