"I am Tini."
When saying "I am Tini", is this not in the sense of "Hi, nice to meet you, I'm Tini."?
In such case, possibly "saya bernama Tini" should be acceptable as well as just "saya tini" which seems somewhat ambiguous.
I thought with Indonesian the subject comes first. So it would be Tini saya. No?
Tini saya means "my Tini". In a noun phrase, the latter word modifies the previous one. Also, you are wrong, the subject here is saya, not Tini.
Although people will still understand you if you say "Tini saya" with an emphasis on "Tini".
I spent a month in Indonesia. It's my understanding that you can say either "Nama saya Tini" or "Saya nama Tini." Either order should be acceptable.
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").
I spent a year in Indonesia, I've always found it to be Nama saya Tini in syntax structure. Truth be told however, there are so many dialects in Indonesian, it may be correct for some groups and wrong for others, regardless Duo Lingo does not have it right here.