"I am Tini."
It's not correct here, even if the meaning is close, as the app expects the literal translation, to check you've understood well the meaning of the words:
My name is Tini = Nama saya Tini.
I am Tini = Saya Tini.
There's 2 ways to say it in English (I am, my name is) and 2 ways to say it in Indonesian, so we have to match the translations with the closer one.
The reason why they make the difference, is also probably because "Nama saya..." is more formal than "Saya..."
Saya bernama Tini = My name is Tini.
With "bernama" a verb meaning to be named.
I agree that the meaning is the same that when you say "I am Tini", but there's 2 ways to say it in English (I am.../My name is...), and 2 ways to say it in Indonesian, so just match the litteral translation, and keep "Nama saya Tini" or "Saya bernama Tini" for "My name is Tini".
Why do you say that "Saya Tini" is ambiguous?
No, "Saya nama Tini" means "I am Tini's name (not Tini itself)". Word order matters in this context. In "Nama saya Tini", nama is tied to saya, forming the noun phrase "my name". While in "Saya nama Tini", nama is tied to Tini, forming the noun phrase "Tini's name".
Perhaps you were thinking of "Saya bernama Tini", in which the prefix ber- verbify the noun (literally it means "I-to have a name-Tini").
There's an official Indonesian (the one you find in Kamus besar: Bahasa Indonesia resmi) and it's the one taught here. They don't mix it with Indonesian dialects, or with slang. Because there's a lot of slang in Indonesian, very often people think there's no official Indonesian language.
It's like with any language, when you learn on a site or in books, you learn the official proper one, and after that, you can learn slang or dialects.
As we are here to learn proper and official Indonesian (dialects are something else):
"Nama saya Tini" makes sense, but not "Saya nama Tini", as it could be something like "I am Tini's name".