When you open a programme and it's 'Icon' shows on the 'Task Bar'
If you 'Right Click' the 'Icon'
In the pop-up menu Click on 'Pin this program to task bar'
It is then pinned.
It should not move unless you un-pin it.
When viewing a website I drag the website icon in the address bar, down into the "Pinned Section" of the above picture,where I can right click the firefox icon in Pic 1 and click on one of the pinned websites I have saved. This works fine until there is an update for Firefox when I loose all my Pinned Items.
I hope this is of help.
The reason I asked which FF you were using is because I use FF6 and your second pop-up menu isn't the same as mine? One thing that I've noticed is when I Right Click on any of the pinned icons on the Task bar, all I get is a pop-up menu showing only things from 'Tasks' down (nothing above?) In other words none of them show a list of pinned objects?
Also are you aware that in the 'Pinned' section FF doesn't feature?