I'm not sure what is being argued by the semantics here but the app store is an important aspect of the device but does that make it not an internet service? An app store by definition is almost a direct equivalent to a web browser. It's just a tool to allow you to move data/code onto your device to do more things. Without it, your device still has a functioning OS. Apple has coupled the functionality of app installation and a curated remote repository of apps on the internet and bundled it as "the app store." I'd say most people aren't referring to the app installation aspect when they refer to the app store.
If an app store is a glorified web browser and apple is maintaining content that is loaded into it, it's almost the definition of an internet service no?