Any documents in your OmniPresence folder on the Mac can be moved into the iCloud Drive section of the Finder. You can drag and drop in the Finder to move them to the desired location, with any folder organization you like (they don’t need to remain in a folder named “OmniPresence”). Doing this will make those documents sync via iCloud Drive, so there would no longer be a need to continue to run the OmniPresence for Mac app at all. After relocating your files to iCloud Drive, OmniPresence can be turned off (App Icon > Gear > Quit OmniPresence), and the app can be uninstalled.
To be crystal clear though — we recommend against syncing any folder using multiple sync services simultaneously, so any files on your Mac that sync via iCloud Drive should not be synced by OmniPresence at the same time.
You can still manually save documents as always in our Mac apps by using File > Save (⌘S). That button you were pressing on iOS did used to trigger a sync, but for saving purposes the iOS apps actually always relied on the operating system which has its own background triggers for autosaves — most commonly, that occurs when an app is backgrounded, or if you exit out of a document.
All file management in iCloud Drive on both Mac and iOS is handled automatically by Apple’s cloud provider. The “conflict files” you mentioned were OmniPresence’s own way of resolving sync conflicts, but iCloud’s sync and conflict resolution methods work very differently, so you may no longer see conflict files like this after moving away from OmniPresence. Hopefully this also means you need to pay less attention to when and how you save documents when working on iOS. Hope this helps!