You are doing this correctly, but what you are missing is the fact that apps are NOT stored in the backup. Apps have to be synced back to the iPad after you restore from the backup. That is how it is supposed to work with iTunes backups.
To be safe, just start over again. Transfer your purchases and backup the iPad again. When you get the replacement, you will want to restore from the backup and then sync with iTunes in order to restore all of your media to the device.
The backup contains app data and device settings, Safari History, notes, calendar entries, your camera roll, etc. but it does not contain the media. The media must be synced back to the iPad.
As a test for .... your own edification, connect the iPad to the computer's iTunes again and look at the apps list and see for yourself that the app that you deleted from the iPad is still in fact in iTunes on your computer. It should be there. Just remember that the app must also be select/checked to sync back to the iPad. If there is not "tick" mark in the box next to the app name, click on the box to select the app again.
read this to see exactly what is stored in the iTunes backup.
http://support.apple.com/kb/HT4946
More helpful reading.
Transfer purchases.
http://support.apple.com/kb/HT1848
How to backup and restore from a backup
http://support.apple.com/kb/HT1766
In an absolute worst case scenario, you can download any purchased apps again for free in the purchased tab of the App Store as long as you use the same Apple ID that you bought them with.
Downloading Past Purchases.
http://support.apple.com/kb/HT2519
You are doing everything that you are supposed to do, so you should be fine.