Present perfect - the action is completed or not completed?

The present perfect is simply the term given to the have + past participle construction in English.

It is true that in many contexts the present perfect refers to what has been 'achieved, finished, completed' (with present relevance). For example:

I've read the book you lent me.

In this context read can be classified (in Quirk's terminology) as conclusive/accomplishment.

However, the present perfect can also be used with verbs that can be classified in their context as nonconclusive/activity, such as live.

She has lived in this city for ten years.

The fact that the Latin word perfectus translates as finished or completed does not mean that the present perfect construction, in modern English, can refer only to contexts where an action has been completed.

Reference: Quirk et al. A Comprehensive Grammar Of The English Language (p201)