Backing up when logged into wrong iTunes account
iCloud and iTunes backups are totally separate. If you backed up to your computer, it doesn't matter whose account was logged in on iTunes. The backup is on your laptop.

iCloud backups occur directly from the device and who is logged into the device does matter in that case.

Also I think you are misunderstanding what happens during a backup. Nothing is transferred to the phone. What you are describing would be a restore.