The cheats given on this site are all correct, but the cheats for invincibility etc, can only be installed from the 'pause menu', and this is only available in the 'training' or specific missions. These specific missions are only opened up when you use the first cheat as shown below. Then during THESE missions you can bring up the 'Pause menu' which allows you to install all the cheats in that mission.
First you have to bring up the Mission window.
From there, you must first activate the cheat that will open up new missions.
This is the Mission select cheat given in most cheat lists:
Hold down together: Left trigger + Right trigger + Back, and then press Y, RB, Y, RB, X. I wrap an elastic band (doubled) around each arm of the controller, and then around each trigger, to hold them down, while I hold down the Back button with my thumb (left of the Start button), then my other hand is free to press the coloured buttons and bumpers. As you press each of the coloured buttons and bumpers, you will hear a click, and when you have fully entered the cheat, you will hear a similar longer sound. This shows that you have entered the cheat successfully, and this will open a number of mission choices.
Now, when you go into any of THESE missions, when they begin, you can press the 'Start' button, which will bring up the Pause Menu, and you can then enter any of the cheats for invincibility, etc.
All of these cheats are on this site, and are easily found.
You will find that with your elastic bands holding down the triggers, once you have completed one cheat, just release the Back button. Then press the Back button down again, and press the coloured buttons and bumpers to key in the next cheat. When you have finished all your cheats, you can remove the elastic bands.
You can now play your mission with all your cheats working!
Hope this helps, it took me an age to work it out! Good luck!