Meanings drift over time. We can probably credit the terms 'REST API', 'HTTP API', and 'JSON API' for popularizing the usage of the term for 'web service', especially since that term went out of fashion in the mid-2000s.
I do understand, however, that meanings drift. Mostly because undereducated and clueless people, unfortunately.
Until it's not disambiguated, especially that API and service implementing the API are in quite close domains and they will cause trouble in understanding. Really, no need to pollute our already difficult jargon with "API" meaning "service" ("API service", if you feel that "service" is overloaded).