Present Perfect Tense

Hi all,

I am confused again about the use of present perfect tense.

Last Friday my British supervisor gave me a task, the deadline was Tuesday. Then he asked me about this task on Monday: "Did you complete it?"

I thought a British guy would say: "Have you completed?" as he is asking "if the document is ready or not now"

Then I talked to some British people and they said either of them is fine.

I expected to hear "Have you completed?" because in my grammar book "English Grammar in Use - Raymond Murphy, Cambridge University" says in an Appendix about the differences between American and British English that:

British: The present perfect is used for an action in the past with a result now;

I have lost my key. Have you seen it?
Sally is not here. She's gone out.
Have you finished your job yet?

American: The present perfect or past simple can be used

I have lost my key. Have you seen it? OR I lost my key did you see it?
Sally is not here. She's gone out. OR She went out.
Have you finished your job yet? OR Did you finish your job yet?

I talked to a British girl about these. She sad "I lost my key" sounds good too, and she said if she see me in my hospital while I am holding my arm she can say "Did you break your arm?" instead of "Have you broken your arm?" as it is easier.

What do you think about these? Especially "Have you finished your job yet / Did you finish your job yet" is very similar to "Have you completed it (yet) / Did you complete it (yet)?"

Is my book wrong as it claims British don't use past simple if they talk about the result of the action or outdated or formally true? Or Do I miss something?

Can you help me please?