[Verizon Samsung Galaxy S6 Edge] Kevin M:
Well I was foolish enough to not disable Verizon Cloud as soon as possible. And it backs up pretty close to everything. Including contacts and texts. This isn't Galaxy S6 specific but I'm posting it here for fellow S6 edge people. Mods, if you'd like this moved elsewhere please do. Here's how to purge your data from the service without having to go on the VZW website:

Photos / Videos / Music / Documents

In the Verizon Cloud app open up the left drawer and go to Settings

For What to back up unselect Photos, Videos, Music, and Documents.

In the left drawer go to All Files

Enter the Mobile directory

Long press on your device's directory to select it. Tap the menu button and click delete.

To verify all four were deleted look at the other sections of the app and they should be empty.

Messages

As far as I can tell this is the only way to delete your backed up messages from the application.

Put your device into Airplane Mode and re-enable WiFi. This is to prevent new messages from getting lost.

In the Verizon Cloud app open up the left drawer and go to Settings

For What to back up make certain Messages is selected.

Use your preferred SMS / MMS backup solution to backup all your messages. Titanium Backup works very well for me. To do this open Titanium Backup and tap Menu. Select Backup data to XML... and once its done just save it locally to your device. I've also had luck on other devices with the SMS Backup & Restore app.

In your SMS app delete every single conversation / thread.

Close the Verizon Cloud app fully for good measure. Open it.

On the home screen of the app you should see something about having "X" number of messages.

Tap the menu button and select Back up now.

Depending on the number of messages you have saved this will take a few minutes. The app appears to go through each message individually saved in the "cloud" and deletes them one by one.

After sync has been completed the app should say "0 messages" are present

Now you may disable message syncing in the Cloud app.

Restore your backup. In Titanium Backup the restore option is in the same menu as the backup option.

Disable Airplane Mode.

Contacts

No luck on this yet, unfortunately. Deleting contacts on your device will work (just as it did with messages) but that will delete your contacts off Google and elsewhere too. :( You'll have to use the web interface. Uncheck "Contacts" syncing in the app before you go deleting anything on the site.

Bonus: How do I access the contacts and Verizon cloud storage website if I am not the primary phone number on the account?

I'll admit this took me hours to figure out the first time. It's completely non-obvious that the primary account cannot access the Cloud data for subaccounts or other numbers. Scenario: Your phone number is part of a family plan or something and it is not the master/primary number associated with the account.

Login to Verizon Wireless as the primary account holder.

Add the number/person who needs access as an "Account Manager." I believe this can be found under the My Profile section.

Logout. You should get a text and email for the number/email specified in the account management step.

Login to this new account with the new number and temporary password.

Setup your profile as requested.

You are now able to access the Cloud data online from this account for this number.

Perhaps there is an easier way to do this without needing to add the other number as an account manager but I do not know of it.

After you've removed everything from Verizon Cloud you can safely remove/disable/freeze it. And hopefully never need to worry about it again until inevitably buy another Verizon phone.