This gets problematic as both are called the same, but are quite different.
Since Drupal 8 doesn't have anymore a module called entity.module, there isn't any conflict problem.
Even in the case entity.module would be re-introduced in a Drupal version, we can still use entity-api for the Entity API introduced in Drupal 7, and for any questions about the entity code present in Drupal (whatever the code is present in entity.inc, or entity.module).
For the Entity API module, there is the entity.module which has the following excerpt:
The Entity API module is a third-party module that extends the Entity API implemented in Drupal 7. For questions about the Entity API, use "entity-api."
The term "API" both refers to the API in core, and also happens to be the name of the contrib module. That is the source of confusion, so to fix the confusion we get rid of the term.