Eve gelince ilk yaptığım şey duş almaktır. ----> Valid

Eve geldiğim zaman ilk yaptığım şey duş almaktır ----> Valid

The two sentences above are not equal. The first one can even be called ambigious.

The second sentence is clear that the speaker takes a shower upon his/her own arrival home.  (that the sentence is preceeded by "Ben" is obvious)

In the first sentence, it is not clear whether the shower follows speaker´s own arrival home - or the shower has to wait until a third party arrives. (Whether a "Ben" or an "O" preceeds this sentence is ambigious).

Wow, this reminds me so much about my first visit to Turkey... incredible. I wish you the best of everything  

/Rob

In the context -as speakers know who they are talking about- it can replace with  -Inda. In our example ;

Just few days are left. I can´t wait to see you... When I come, I want to talk with you about something. And don´t ask me about what, I won´t tell you now.

[Oraya] geldiğimde, seninle konuşmak istiyorum = [Oraya] gelince seninle konuşmak istiyorum.

Therefore two sentences above are equal as they refer the same person [I]

Eve gelince ilk yaptığın şey nedir ? [ what is the first thing you do when you come home?]

Eve gelince ilk yaptığım şey duş almaktır.

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Eve geldiğinde ilk yaptığın şey nedir ? [ what is the first thing you do when you come home?]

Eve geldiğimde ilk yaptığım şey duş almaktır.

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Above , two equal [in meaning] answer to two equal [in meaning] questions.

Again,we, as speakers normally know the context while talking ,they both are exactly the same. Unless you pick them out of context and compare them with eachother then we might get it wrong.

Learners know that -ince takes its subject from the main clause predicate (unless not mentioned in the gerund).

The main clause here is a nominal clause,

                                      İlk şey...duş almaktır

so predicate must be found closer now, it is yaptığım of the relative clause. The native comments show here that it is not a matter-of-cause that -ince can adapt its person from a participle. Context is needed to avoid the trap of ambiguity.

If it is a general statement without a context as below, go for the person that is in main clause predicate.

Türkçe konuşunca kendimi daha iyi hissediyorum ---> I feel better when I speak Turkish.

Otherwise, we normally put person when we refer someone else.

Ayşe Türkçe konuşunca kendimi daha iyi hissediyorum --> I feel better when Ayşe speaks Turkish.

There is no need for long crappy discussions.

EVE GELİNCE İLK YAPTIĞIM ŞEY DUŞ ALMAKTIR, as it is, is an ambigious sentence which should technically be avoided in decent Turkish.

It is not clear which one of the following the speaker wishes to declare,

- Sevgilim eve gelince ilk yaptığım şey duş almaktır.

- Ben eve gelince ilk yaptığım şey duş almaktır.

You cant discuss anything without using crappy words ,can you ?

"Your so called technically be avoided form" is well used in daily life Turkish my friend.

We dont waste our breath by putting "Ben" when we refer ourselves. And when we refer somebody else we surely mention that person. [in your case " Sevgilim"] So Turks are clever enough when they talk.

While communicating we do not always eliminate every possibility for misunderstanding. In a dialogue speakers refer to their common knowledge all the time.

Ambiguity is a part of human language. It is necessary also. Where else would poets and liars find their inspiration?

Yes, poets´  intentional enigmatic ambiguousness is sweet  whereas liar´s  bitter but necessary too.

More crap !