If G-Mail works the same as my using Hotmail, which I believe it does, Verizon Cloud will back up Verizon related items; gallery photos, gallery videos, Message+ text threads, Contacts located on "Phone" etc. It won't back up anything 3rd party related.
You could use Google Drive, though I am not familiar with it, it may back up contacts located both in Phone and that are linked via your G-Mail account.
What's your ultimate goal?
It sounds like you should be doing things one way or the other, but that you're doing it 2 ways - like myself. Occasionally I sit and go through my contacts and add them into either my personal Hotmail and my work hotmail and delete them from the Cloud. I find it easier to have them sync to my phone from my e-mail because I can easily use the computer to add notes and make changes and they sync quickly.