I call that form as Narrative perfective tense. In Turkish -mIştı  marking is used for actions that are completed in the past. By using - mIştı , we are ;

1. narrating the past action to someone

2. I witnessed or I myself  performed that action.

3. The action occurred in the definite past *

As in your example; 

Ben size söz vermiştim. ---> I had promised you.

1. I am reminding you, narrating you the event [my promising you].

2. I remember and i witnessed the action that it has been performed by me.

3. It occurred in the past and it occurred in a definite time in the past.

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The only distinctive difference between simple past tense [size söz verdim] and [size söz vermiştim] is ,  we are narrating the past event that we witnessed.

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3. 3. The action occurred in the definite past *

Although this form technically should indicate the indefinite past because of tense narrowing at present time it is used for definite past.

 According to Prof.Hamza Zülfikar ;

The definite perfective past    -dıydı    is replaced by  indefinite perfective past - mıştı 

söz verdiydim ---------------------------------> söz vermiştim

                           tense narrowed into

- as a result , at present time, we use -mişti form for definite perfective past time.

The nearest tense in English is  plu-perfect tense [had]

Now I understand why Turks sometimes use English past perfect structure in a strange way. They imitate this narrative!

Second language speakers often try to reorganize the language they are learning.

I wonder what would be the exact English translation for Ben size söz vermiştim, including the connotations tunci described. Maybe they would be expressed with intonation.

Yes, I would stress the ´had´ part of the sentence in English

I had promised you, ....... and I will try to keep my promise.

(I am aware I made a promise to you in the past)

I finally read this. I didn´t have enough time..Sağ ol tunci,çooooooook. Anlıyorum şimdi.