It clearly states on the website under Support:
Can I move my broadband to another premises?
Broadband is attached to a physical line so you would need to request a cease for your broadband and then place an order for a new broadband service at the new premises.
To request a cease you would contact us via email or support ticket. Please remember to include the phone number the broadband is on, the address and post code and the date you wish the cease to commence.
If you wish to place a new order with us for the new premises please include the address and post code and the phone number for the new premises as well as the date we should place the order.
For the confusion around the new property Vivaciti have made the OP an offer due to the confusion around price by the looks of it.
The process to move is clearly stated in their support pages. So I do not see why Vivaciti should waive the cease fee for the old property.
Based on that it leaves the OP having to £9.60 (new property) + ADSL Cease charge £31.12 (old property) = £40.72, not sure if the cease fee includes VAT or not.
At this point the OP should actually take the offer as there is blame on both sides, the OP should actually read T&Cs and the support pages going forward, Vivaciti need to improve their comms and outline the move process and the associated costs.
Vivaciti have made a fair offer if that is the offer that has been made. It would cost more than that to have another line activated!
There does seem to be selective posting of what has been said. The OP is under less obligation to keep certain information private as he is not covered by the data protection act...
Fibre > Sky Fibre Pro > Pulse8 Fibre XL - 14ms Ping, Sync ~ 65.78/18.73Mbps - BQM