How do I replace the old contacts backed up on iCloud with a new set of contacts I've just created on my iPhone?

Got it. To replace the contacts in iCloud with the ones on your phone do the following:

1. Go to iCloud.com from your computer, log into your iCloud account and open Contacts, click on All Contacts on the left, to the right click on any single contact to select it, click on the Actions (gear shaped) icon at the bottom left and choose Select All, then press the Delete key on your keyboard.
2. As a precaution, download the app My Contacts Backup. (If you have more than 500 contacts to transfer you will need the “Pro” version.) Use the app to back up the remaining contacts on your phone (from the other accounts) as a vCard attachment to an email that you send to yourself. Confirm that you have received the email.
3. Go to Settings>iCloud on your phone, turn Contacts on and choose Merge when prompted. Check iCloud.com to confirm that they are all in iCloud. If for some reason they aren't, and they are no longer on your phone, open the My Contacts Backup email you sent earlier and tap the attachment to import them to iCloud.

Is your phone syncing contacts with iCloud right now? (In other words, do you have contacts turned on in Settings>iCloud on your phone?)

You're welcome.