Unfortunately there is no standard way of recording game footage from games consoles without extra hardware.
Some games have video capture tools built in (like Halo Reach), but if they allow access to those videos outside of the console ecosystem it can be in many different ways. For your Halo Reach example, you actually require an account on http://www.Bungie.net. Once this account is linked to your Xbox LIVE gamertag, you can access your fileshare from the Profile->File Share option on the website and access any videos you have saved.
As this functionality is Bungie / Halo Reach specific, this won't necessarily work the same for Modern Warfare 2 (or any other games). So this is where a specific piece of hardware; either a capture card that slots into your PC or an external capture unit such as this one that I know games testers working on real games use. Check out http://www.hauppauge.co.uk/site/products/data_hdpvr.html.
Obviously specialised hardware can have quite a high cost associated with it, but it does allow you to record every part of gameplay as you see fit and would work with any games console, not just Xbox 360.