IaaS vs. PaaS — Confusion

I was recently involved in an opportunity where I had to assess integration platforms for a customer. The requirements were broad and typical of most customers. I included the popular iPaaS (Integration Platform as a Service) platforms and learnt that the customer was interested in looking at PaaS (Platform as a Service) options too. But some of the products that I was evaluating were being positioned as IaaS (Infrastructure as a Service).

Now, as I understand, IaaS is concerned about providing Virtual Machines and Cloud resources, where we install and manage our own platform , and PaaS means availability of a platform that is managed by the Cloud provider. The use of the term IaaS for integration platforms baffled me and I was not able to get a satisfactory response from the respective SME on why it was termed so.

Have you come across such usage? Or is my understanding of what an IaaS or PaaS is incorrect? Please let me know your thoughts.

Cross posted at https://www.linkedin.com/pulse/iaas-vs-paas-confusion-danesh-hussain-zaki/