Based on my understanding, Operating System in considered as PaaS(Platform As Service). It is a category of cloud computing that provides a platform and environment to allow developers to build applications and services over the internet.It also allows users to create software applications using tools supplied by the provider.
It depends on what you mean by server. If it is the OS that runs your virtualization software it is acting only as support for the IaaS, albeit not really IaaS per se. If by server you mean a VM or physical machine with an OS that you or the cloud provider run on the cloud to handle your apps without you worrying about VM management then it is considered part of the PaaS. SaaS is a level up and consists in accessing directly the app completely hiding the OS.
On the one hand, according to the NIST definition of Cloud Computing (the most widely accepted on, “The NIST Definition of Cloud Computing , “Special Publication 800-145”, http://csrc.nist.gov/publications/nistpubs/800-145/SP800-145.pdf) and quoting from it: “IaaS: The capability provided to the consumer is to provision processing, storage, network and other fundamental computing resources where the consumer is able to able to deploy and run arbitrary software, wick can include operating system and applications. The consumer does not manage or control the underlying cloud infrastructure, bur has control over operating systems and deployed applications …”. So, puristically speaking, the Operating System is not part of IaaS, as is stated by Thomas Pasquier.
On the other hand, in the practice some Cloud Providers (as Shiping Chen suggests), in their IaaS provision dashboards let you chose the operating system (“image”) to deploy in the Virtual Machine (VM) you provision. So they are responsible of guarantee that “image” is good; so in some way they have a partial responsibility on the Operating System level (crossing the border of he IaaS) but it’s only in the first deployment of the operating system in the VM; after that you can parameterize the operating System so you’re full responsible of the Operating System and software above. Note, of course, other IaaS cloud providers let you to upload you own Operating System images, so they are responsible for providing you the VM on the hypervisor (or container) chosen by them, but nothing else, matching the purist definition of IaaS.
Basically in simple terms IAAS is a compute fabric think hardware, hypervisor, compute resources like network bandwidth, SAN storage, CPU and RAM resources. In a nutshell can I provision to it? resource from it? or allocate to it? I know it's a multifaceted question and depending on the service provider a somewhat vague term but you should get the general idea.