The paid app binary and the non paid binary are exactly the same. You do not download a new apk when you purchase content in an app. Sync's "online verification" is through Google Play' libraries. When you purchase, your device received an entitlement to that specific purchase which allows it to re-verify that purchase without hitting Google's services each time the app is opened. Since it was a lifetime subscription, this entitlement never had to be renewed.

As for it working between phones, either the entitlement is bundled with the app (which I'm not positive of), or it's entirely possible the version you had was bugged so that it acted as though you had purchased even though you had not in the event these libraries were not available.

Either way makes no difference. You need the entitlement to activate the features, not an entirely new APK.