I always thought "protocol" was a more appropriate term for such internet-based APIs. SMTP and HTTP are clearly protocols, and even APIs built on top of HTTP are generally referred to as protocols in the standards world. For example WebDAV (RFC4918) and AtomPub (RFC5023) are both considered protocols.
It seems to be the accepted use these days. I disagree with that use but I don't have a better term and don't want to add to the confusion. But you're absolutely right that an exposed service and an API are different in many fundamental respects. I think the confusion is rooted in the gray area called remote procedure calls, if you implement your service exposure as a remote procedure call (and many of the early ones were done in just that fashion) then the difference is almost moot.
Maybe WSI would be a better moniker? (Web Service Interface)?
It always comes to my mind operating system API when I read such titles.