contact: is a tag namespace prefix, it's not a qualifier for the website tag. Basically they are identical tags, you can read about that here: https://wiki.openstreetmap.org/wiki/Key:contact:website

What you want to do would be best done using the website: prefix (https://wiki.openstreetmap.org/wiki/Key:website:*), so probably website:contact, but there's not a lot of support for that. 