How to close (shutdown/kill) the Android web browser? I did read this question 
What is the right way to close apps in Android? but I want to know how to close - that means close - application to save power and download limit.

My observations tell me that Back button brings me to the previous page (something I am not after) while Home seems to me leaves the page loading in background. So the question is how do I exit the browser.

I tried Advanced Application Killer but I don't think it worked ...

Using 
HTC Desire on Android 2.1
Closing the browser has no benefit. The android subsystem manages the memory on the device really well and will free up memory from background applications if it needs it. As I understand it, it does this by saving the application state in a database on disk. If you run the application again, it can then rebuild its state from the info in the database.

If you want to stop the browser from downloading and you can't see the "Stop" button, press the menu button. This will bring up the address bar allowing you to press the stop button. Although if the address bar has disappeared then chances are your phone is not downloading anything, unless the site you are on has Javascript on it which is causing it to refresh elements regularly. If this is the case you are on your own.