slowdown Should I update it or should I just updated it from GrapheneOS' App Store?
If I update it from Play Store it would be Google's version and hence maybe it would have more privileges than the from GrapheneOS' App Store?
No, the versions of Play Store, Play Services and Google Services Framework that you get through the App Store are the unmodified versions that are made by Google. The compatibility layer GmsCompatConfig instructs Play Services, Play Store and GSF how to work without privileged access – in the Android app sandbox – no matter if you install them from App Store or Play Store. Please see https://grapheneos.org/usage#sandboxed-google-play
You can update Play Services and Play Store from either App Store or Play Store, but the Play Store is usually some versions ahead of App Store.
The notification in your first screenshot ("It's recommended to allow this") is delivered by GrapheneOS and not the Play Store. It's perfectly safe to let Play Store update itself and Play Services.