You may need to get a better sense of the evolvement of what we currently call "Tachanun" and what Nefilat Apai'im refers to, as they are not the same thing.
I'm excerpting various ideas from several articles that were compiled into a single Beurei Hatefilah newsletter. Since this is in PDF. I may have trouble copying and pasting some sections, so I'll refer you to the page(s). The page numbers are the PHYSICAL pg. numbers in the PDF, not the page number written in the page header.
pg. 29 - Three rites follow Shemnoeh Esreh - Viduy, Tachanun &
He seems to emphasize that it must be in this order.
Then he focuses on the 1st part of Tachanun:
The opening verse Vayomer David was not always said & does not appear in early Siddurim. Aruch Hashulchan claims that it was not their minhag to say that line.
The 2nd line - Rachum Vechanun was composed by Chaza"l & that is viduy
The 3rd part containing Tehillim 6 is Tachanun.
As there are various orders between Nusach Ashkenaz & Sefard, I will discuss Nusach Ashkenaz, here. (See somewhere in the newsletter some mention of other customs that mention different paragraphs said other than Rachum V'chanun (Tehillim 6).)
p. 31 (translation of Siddur MH"R Shabtai Sofer):
The practice to recite the line: Rachum V’Chanun Chatasi before
reciting the chapter of Tehilim is based on what is written in the
Zohar, Parshas Pikeudei, page 472 in the large print edition and on
page 262 side one in the small print edition, that one should confess
his sins after finishing Shemona Esrei and before falling on one’s
face. That is what the To’Eles Yaakov wrote on page 33 side 1. A
similar point is raised by the book: Seder Ha’Yom page 20 side 1. The
key element of confession is the words: but we have sinned as it is
written at the end of the last chapter of Maseches Yoma page 87 side
1. It is further written there that confession must be recited while standing. The same point is made in the Zohar, ibid. So too the
Schulchan Aruch held in Orach Chaim Siman 607 paragraph 3. It
therefore appears to me that one should say the words: Rachum V’Chanun
Chatasi, which represents the essence of the confession, while in a
standing position. Then one should sit down, lean one’s head to the
left and then recite the chapter of Tehillim.
So, it seems that the REAL part of Nefillat Apa'im is only Tehillim 6 and that's the only part when you should lean your head! An important point to learn from this excerpt - Don't lean for either Vayomer David or Rachum V'Chanun. (And to think that I've been doing it wrong all these years!)
Begining on p. 58, you will find a discussion of Hashem Elokei Yisra'el. One notable points in that section:
Machzor Vitri says that this Piyut was originally said responsively by the chazzan and cong. That's no longer done. But, from the fact of its original planning, it seems that this was NOT part of Nefilat Apa'im, otherwise the Chazan wouldn't be standing. (While not explicitly stated, there, I feel safe to assume that any responsive reading by the chazzan would not be done with him leaning.)
There are other points in the newsletter that I encourage you to read, as they will offer different angles to the structure of what we now call Tachanun.