Names:
On the desktop there are two apps, similar in concept but significantly different:
1. Lightroom Classic.....the traditional app which stores and manages images on local disk drives, initially available with a perpetual license (versions 1 through to 6), before it was renamed as Lightroom Classic CC (version 7, it's now at version 10) and is only available with a subscription. For a very short time during the Lightroom 5/6 cycles, it was available both via perpetual license or via subscription. Only during the Lightroom 6 cycle was the subscription version known as LRCC 2015.xx.
2. Lightroom.....at the same time as Lightroom Classic CC was released, a new desktop app was also released, which is basically the desktop version of the Lightroom Mobile app, i.e. images are stored in the cloud. Initially is was named as Lightroom CC (aka LRCC), but a year or so later the "CC" was dropped from almost all Adobe applications, so on the desktop we now have Lightroom Classic and Lightroom. You may see Lightroom sometimes referred to as Lightroom "cloudy", but that should be no longer necessary. Stick with the actual names and it will be fine. Too many users still use LRCC, which as you see was previously used by BOTH the desktop apps....so it's the continued community use of LRCC which causes most confusion these days.
Lightroom Mobile....is the generic name for the Lightroom apps which are available for "mobile" devices, i.e. smartphones and tablets. The apps are more or less identical in functionality (though there are still a few differences), but need to be different based on the operating system and the hardware. So for instance there's a specific version for iPhones, another specific version for iPads, a specific version for all Android devices, and another version specific to Apple TV. The individual apps may have different names, such as Photo Editor on iOS, but nobody (not even Adobe) uses those specific names, everyone uses the term Lightroom Mobile (LrMobile) and then adds the specifics about the OS and if needed the device type.
Regarding the "best approach" for syncing iPhone photos to LrClassic, I answered a similar question a few days ago:
Take Lightroom on a trip