Did you sign in to iCloud on your new phone? From your original description, your Mac was definitely syncing with iCloud. You indicated that you opened iCloud in System Preferences, and the various boxes (Contacts, Notes, Calendars, Reminders, etc.) were checked. This means that you Mac was syncing that data with iCloud.
An iCloud backup is an entirely different thing. An iCloud backup is done only for your devices (iPhone, iPad, iPod). When you create a backup of one of your devices, it backs up what you have selected for it to back up (on the device, not the Mac). Backups are used when you need to restore your device after it has been erased for one reason or another, or to set up a new device, if you do not want to manually download your apps and media, and you want the Settings on the new device to be identical to the Settings on the old device.
iCloud sync is a real-time ongoing process that syncs any data housed on your device or Mac (that are signed into the same iCloud account) to iCloud. So, iCloud is the "holding facility" for that information. iCloud then pushes that info to each device or Mac that is signed in to that same iCloud account and that has those same options selected for sync.
So, when you unchecked "Calendars" it said it would remove the Calendars from your Mac, because those Calendars are housed in iCloud and sync'd to the Mac from iCloud. Once you re-checked it, they were all pushed back to the Mac.
You can see the contents of your iCloud information by going to iCloud.com on your Mac, and sign in with the Apple/iCloud ID that is signed into your iCloud account. You can then click on any of the options (Contacts, Calendars, Notes, etc.) to see what is in iCloud for those items.
If you do not sign into iCloud on your device, then you would be setting up all of that infor on the device only, and you would retrieve it by doing a backup to iTunes, making sure that the Settings tab for your device in iTunes did not have iCloud checked for those items.
But the easiest thing to do is to set up the iCloud sync. That way, when you add a contact to your Mac, it shows up on your phone and vice versa. When you sign out of iCloud on your device or Mac, it does say that it will delete that info from the device or Mac that you are signing out of, but it is retained in iCloud. So the next time you sign back in to iCloud on that device or Mac, it is all pushed back from iCloud.
Best of luck,
GB