Or as the service provider I use calls it "cloud hosting".

Without getting into a buzzword fight about what the cloud is, in my experience what almost 100% of customers are really saying when they ask for the product to be in the cloud is for it to work on their phone/tablet/browser without requiring installation (or at the most requiring installation from an app store). The stuff happening on the other side of the browser/app is of absolutely no interest to them (so long as it is secure, backed up, and highly available). Whether it's a real McCoy "cloud service" like Azure or "cloud hosted" VPS is neither here nor there for the customer - it's an implementation choice for the vendor (as everything on that side of the equation is up to the vendor to build, maintain, and pay for).

Their other customer concern is the cost, and in my experience the less complicated the better. This makes the choice of "cloud service" or "cloud hosting" a marketing issue too. I've found that it is easier to sell "get access to our service for $X/month" than "get access to our service, for $X+$Y+$Z per month for CPU/IO/Network bandwidth - just remember to check the dashboard periodically and be careful how much you are using if you don't want a surprising invoice at the end of the month!".

That's my experience with my product/service & market - depending on your product/service and market, your experience might be the completely different of course.