Read it carefully " over which the holder has lawful authority to shoot. ", it does not specify any land by name so if a farmer gives you permission to shoot his land, you can. It's open.
With regard to the management of any other wildlife etc, it would seem that the first clause that specifies which rifle you can use for what, is at odds with the ammunition clause which would suggest that it includes fox.
I would still talk to my FLO and get the deer rifle authorised for fox. ACPO, I believe it was, have agreed this so there should be no problem.
John