Who owns the legal partenity of a website, the designer or

I contacted with a web designer to redesign our existing website and only when it is completed will they upload the new site. I had to pay £1800 including vat £900 up front and the balance on completion. I signed a contract that states that the company "may include the statement "designed by abc or site designed by abc" together with a link to teir website.
Although that is the case that I signed the contract I did not agree that they should and upto the final delivery of the website we stated that we did not want to carry a free advert and that we would charge them.
However, on the 26th August more than 6 months late of delivery we were told to pay the final balance and then our new site will be uploaded to our hosting server. Yes we saw the final production and we did not have enough time to fully test the product. On that day after payment the site did not display and the designer were away on holiday within hours of our payment. We know this because they told us they were going away. We strugled to get help fro them by email and for 3 days we had no website. We cotacted another designer who helped us to correct the upload and fixed some issues and so we asked them to remove the "designed by abc" from our site. 2 weeks on their reuturn after we have fixed and continue to find problems with the site they now asked us to put back the credits or we will be charged £3000. We said the site originally was not fit for the purpose but the issued a county court case against us due to be heared in Feb 8 2016
We belive that the copy right law doe not include work created by computer as we provided all the content and pictures and the site is hoted by us and we helped design the layout. what is our position.

Its included in my answer above. It does not say we will have to pay any money if we did not accept the credits to their name.

I dont mind a live conversation but I dont want this to be an authorisation for you to continue charging my card for any additional payments. please confirm that this will not autorise a recuring payment.

It says abc may include the statement "designed by abc or site designed by abc" on each page of the site in a form to be agreed

no we are being sued for £3000 + vat to release the copy right to us. which is my question who owns the copy right of our site.

yes it says under intellectual property rights that third part works means the works and meterials comprised in the site, the intellectual property rigts in which are owned by third party; and abc ltd

Can I scan it to you

I have sent it

Yes what does the Exceptions to the right under section 79 of the copyright, design and patents mean for us in this case.

this is the section I mean

We have since then used another designer to redesign the sight as a result of hacking of the original site with the continued fixes, the site is still not working and is current offline.

this is helpfull as we did send them email that the site had failed to be published and they did not fix it or offer to fix it for 3 days before we fixed it.

Will you be interested inthe case? the hearing is on the 8th Feb 2016 In Croydon. What will you charge?

can you suggest another

thanks