You've hit the nail on the head. Moving your legacy J2EE accounting service running atop Oracle from your server cabinet to a rented virtual server is not cloud. Cloud is about replacing that application entirely and buying it as a service. It's about cutting out the cost of running a server in every office and having an IT guy looking after it.
"What if we get an Internet connection issue and can't pay our bills", you cry. You have a plan, just like you should have a plan for the eventuality of a power cut. Turning the question on its head, what if you have a burglary or a fire? Why should your business be dependent on the physical integrity of your office? I'll rather trust my core business to a collection of professionally run datacenters across the continent.
Many core premises of onsite IT are being eroded.
1) Connectivity is improving. With unmetered 4G being the norm in progressive countries, you have the situation where your phone has a faster internet connection than the office LAN.
2) Geography is less important. Not everyone is working in the same office. Increasingly they may be on the road, working from home, or contracting from another country.
3) Online services are competing with in local services. Those services you are running are being replaced by cheaper alternatives like Gmail, Office 365, Azure Active Directory etc.
Obviously the IT guy will have a hard time accepting the state of things, but it's the way we're going. Deal with it, and adapt.