What's a job title for someone in Marketing & Communications that also manages a web site (design)?

I think the answer lies part in what you do most and part in how your company structures its titles. It seems to me that the lion's share of what you do is internet and technology based, hence I would say Webmaster, or e-Marketing Manager (or Director or Supervisor depending on your company's nomenclature. If you are really doing a lot of broader marketing and a lot of your solutions just happen to use technology, then I think you could push for a change to Marketing Manager (or Director or Supervisor, etc.)