Question: One of the "akin" examples is "Akó ang gumanáp na pi

No. Some speakers use the "aking" class pronouns more often than others, but in general it is a stylistic choice. Aking sounds more formal to a central (manila) or northern Tagalog and poetic. Aking (really just a prepositive akin with the linker) is more common in Southern Tagalog and among some speakers of regional languages of which their native language may prefer the prepositive forms.

One is more "literary", the other is more "casual".Think of it as "my house" and "house of mine".

I agree with answers, it's all about being formal or literary when you say "aking".
"Aking" is the formal/literary version and "ko" is the informal/casual version.
Both means my.

@blueshell Thanks...so how about "ng pulutong sarili ko"

If aking is more formal...would one tend to favor its use in response to a question...as more formal?

@BoraMac Hi dear! Can you please tell me the possible full sentence? "Ng pulutong sarili ko" is actually vague. If I would rearrange, it would be "sarili kong pulutong".I'm sorry for confusing you in any case but allow me to rephrase my answer. Bituingmaykinang is right, "aking" is literary. In conversational use, even if you'll talk to a teacher, older people or even to the President, we would casually use "ko" instead of "aking". However, if you really intend to be formal and polite, you may use it like this:
--> sa akin pong palagay, sa akin pong pulutong (instead of "sa pulutong ko po" or "sa palagay ko po")Otherwise, it depends on the context and intentions. I speak literary Tagalog with my friends when we intend to sound formal or use "malalim (early or old version/deep) na Tagalog". I would use "aking" instead of "ko". Also, when you're in a strict Tagalog class and you're not allowed to speak Taglish (I personally experienced this during my high school), feel free to use "aking" instead of "ko", especially if the teacher likes being "makata" or poetic. I hope this helps you.

@stevesmi
Hello, I hope what I will say now will somehow make things clear.The pronoun "akin" also follows rules. We can use the pronoun "akin" when the noun we want to replace start with "sa/Kay". And "Akin" is in first person point of view.Example o halimbawa:Ang sa bata ay itago mo muna
-> Ang sa akin ay itago mo muna. (If you were the child being referred)
-> Ang sa kanya ay itago mo muna.
(If you are referring to the other child).AnotherAng Kay Nene ay gayon din.
-> Ang sa akin ay gayon din ( if you are Nene)
-> Ang sa kanya ay gayon din. So with your sentence, it is better say "Ako ang gumanap bilang pinuno sa aking pulutong"

@stevesmi
Aside from formal vs informal, one subtle issue here is about emphasis. Like, if you want to emphaize who owns it "akin" is better to use, but if you want to emphasize the object, then say the object first before the pronoun. Like, "Sa aking pera yan." This is MY money.
VS
"Pera ko yan." This MONEY is mine.