I understand from your analyse, gokuyum, that the difference is something like this (if we try sentences which are as close to one another as possible):

1) Ben [başbakanın daha sakin olmasını] rica ediyorum.

2) Ben başbakandan [daha sakin olmasını] rica ediyorum.

The hidden or visible indirect (ablative) object always belongs to the main clause, doesn’t it? If so, how does supposing it there or out of there take us any further? My idea was that in the previous example the governed clause is inlayed as a whole while in the latter the inlayed subject would actually be o/kendisi which means that the inlayed sentences would be something like

1) Başbakan daha sakin olacak.

2) O/kendisi daha sakin olacak.

Not that it matters so much to me. Actually I was just trying to translate something, got stuck and began to play with this thought.

Yes, I think like you.

Here is my addition to the thread and I hope you find it useful.. 

Biz, onun bize gelmesini beklemiyorduk  (onun bize gelmesini => direct object )

Onun bize getirdiği hediye çok hoştu (onun bize getirdiği hediye => subject)

Biz, o bize geldiğinde yemek yiyorduk ( o bize geldiğinde => adverb/adverbial adjunct)

Let´s simplify the sentences (their meanings remain the same):

Gelmesini beklemiyorduk (gelmesini => direct object )

Getirdiği hediye çok hoştu (getirdiği hediye => subject )

Geldiğinde yemek yiyorduk (geldiğinde =>adverb/adverbial adjunct )

I was staring at my own quote for a while before I understood what I maybe must have ment with it. It seems that you found a real pearl, scalpel.

What comes to the inlayed subject the case with personal pnonouns is the simplest of all, of course, as your examples show because as a rule the person becomes visible in the possessive suffix of the predicate.

I must have been thinking about a case where

- predicate is in passive voice

- there is a logical object in the sentence which I have to rise up as the subject

- this object cannot be omitted for some reason (it probably refers to a thing, a state of affairs)

anyway something that I don´t meet every day. That´s why I was asking. I can add the sentence later if I can recall it. And the same nominative - genetive rule held just like si++ answered here.

Let´s keep playing a little more  

x-den y-ı rica et(mek)

Onlar, senden gitmeni rica ediyorlar

Sen, ondan kalmasını rica ettin

O, sizden sessiz olmanızı rica etmişti

Ben, onlardan kayıtsız kalmamalarını rica edeceğim

Both pronouns can be dropped as the personal endings in the DO and predicate are clear enough to know what is the subject and if the IO is 2nd or 3rd person.. 

Gitmeni rica ediyorlar (2nd per. sing. - 3rd per.plu. ) 

Kalmasını rica ettin (3rd per.sing - 2nd per.sing. )

Sessiz olmanızı rica etmişti (2nd per.plu. - 3rd per.sing. )

Kayıtsız kalmamalarını rica edeceğim (3rd per.plu. - 1st per.sing)

I just noticed now that DO is always in me/ma form of verb.

As for your examples.. (nr 2 ) is correct but (nr 1 ) is not

2) Başbakandan (daha sakin olmasını  rica ediyorum. (correct)

1) Ben (başbakanın daha sakin olmasını ) rica ediyorum (incorrect)

It is incorrect because the question "from who" is unanswered..

You can say 

Ben (başbakanın daha sakin olmasını ) kendisinden rica ediyorum

But it doesn´t sound great. 

What? Now I am all confused. Time out. I will have to do some googleing first thing in the morning. I will bring you twenty incorrect sentences, scalpel.

Twenty sentences...sigh...I shouldn´t post anything at midnight. I hope you have a skin thick enough, scalpel.

What you say is very clear. It seems that as a learner I cannot fully understand what Turkish pronoun dropping leads into and that´s why I interpret sentences incorrectly. The length of this thread shows that the issue of inlayed subject has unconsciously worried me since very early stages of learning. It´s partly because sentences in which personal pronouns are used are so simplified that you can´t use them as a model for sentences with noun subjects in spite of their frequence in use.

Impersonal sentences are another source of confusion:

İsteyen tüm arkadaşların katılmasını rica ederim.

There is ø in the place of the indirect object and the genitive noun is the subject of the passive infinitive. Well, it is not the simplest thing in the world.

What comes to structures with rica etmek we are under the wrong headline. The complement clause is not fully inlayed but it stays as a part of the main clause. If it was and inlayed clause our choise for the subject case would be between nominative and genitive.

Why do you think so? I don´t think it is incorrect. It is just the other way of expressing it. It is not necessary that "from who" to be anwered.

I don´t think there is a problem with this sentence.