Is FaaS required to have an official "Cloud"?
Google’s free computing course says:


  The US National Institute of Standards and Technology created a
  definition of cloud, although, there is nothing US specific about it.
  Here it is, cloud computing is a way of using I.T. that has these five
  equally important traits. First, you get computing resources on-demand
  and self-service. All you have to do is use a simple interface and you
  get the processing power, storage, and network you need, with no need
  for human intervention. Second, you access these resources over the
  net from anywhere you want. Third, the provider of those resources has
  a big pool of them and allocates them to customers out of that pool.
  That allows the provider to get economies of scale by buying in bulk
  and pass the savings on to the customers. Customers don't have to know
  or care about the exact physical location of those resources. Fourth,
  the resources are elastic. If you need more resources you can get
  more, rapidly. If you need less, you can scale back. And last, the
  customers pay only for what they use or reserve as they go. If they
  stop using resources, they stop paying.


This seems to be taken from the document at https://www.nist.gov/news-events/news/2011/10/final-version-nist-cloud-computing-definition-published

Using DevOps software with virtual machines, with or without containers, can happen without using cloud. In a similar manner, you can open a corporate account on a public cloud and not do DevOps at all. This happens more than we might expect when big companies are migrating legacy systems into the cloud where they simply have sysadmins or other Ops folks use the web interfaces to the cloud IaaS to manually commission VMs that they then manually configure. I have met teams that are convinced that they are doing DevOps when they are simply doing Ops with a few specialist Cloud Engineers who handle setting Virtual Private Cloud (VPS) meaning private connectivity into the cloud (e.g, VPN or a dedicated connection) and IAM (Identity and Access Management). 

IMHO cloud and devops are most definitely not the same thing but are at their best when combined.