If the thing is not runing on a user computer, it's not local. If a datacenter is down the hall it's still a remote machine. It's always remote, unless the user is physicaly in a datacenter and using the server directly.

This whole argument is proof enough that the original language is confusing. For your argument to make sense we need to assume that local is used as a non-IT term and cloud as an IT term in the same sentence. That is very confusing language no matter how you look at.