Because the meaning is different.
"Non era mia intenzione" means that I would want to not have did/said a specific thing that I did. Often it is a formula to beg pardon. e.g. If accidentally I offend you, I'll say you "scusa, non era mia intenzione" (sorry, it was not my intention - I don't know if this translation makes sense in English). Also can be that your actions generated unexpected (and not desired) things, and than you say that phrase. Therefore "Non era mia intenzione" simply means "non volevo" (I didn't want).
On the other hand, if I say "Non era la mia intenzione" (probably preceded from "questa"), that means that you didn't understand what was my purpose. "Non era la mia intenzione" means exactly "it wasn't my purpose".
I hope I helped...
Sorry, I still fail to see the distinction you're making. Saying exactly "it wasn't my purpose" in English would be the same as saying "it wasn't my intention." Both could be used as apologies.
I completely am missing your point in the second paragraph. Why would 'you' not understand 'my' purpose, when 'I'M' the one saying it wasn't my purpose?
Inserting "la" in "Non era mia intenzione" is wrong, but so (it seems) is deleting it in "Era la mia intenzione dall'inizio". Comparing all the sentences in this lesson, it looks like you should leave "la" out in negative sentences, but have to use it in positive ones. "Non ho intenzione ..." vs "Ho l'intenzione ..." Pretty subtle, quelli Italiani. Ma non sono sicuro se `e vero.
Perhaps it's following the general rule that possessives after the verb drop the article, but usually when there's a noun (not a pronoun) subject before the verb. Having the noun and possessive after the verb doesn't obviate the omission of the article. For example:
La mia casa era bianca. La bianca casa era mia. era mia bianca casa.
I'm really guessing on this one, though.
You should read my comment above.
For your examples, you can't put words how you want. You can say:
1) "La mia casa era bianca" and 2) "La casa bianca era mia" / "Era mia la casa bianca".
1 and 2 have a different meaning.
"La bianca casa era mia" is poetical and/or archaic.
"Era mia bianca casa" is wrong.
You can also say: 3) "la mia casa era la bianca" and 4) "la casa bianca era la mia". And they all have a different meaning.
I meant to type "era mia casa bianca" if that makes any difference.
I don't think meaning is important here - it's the question of where you omit the article and where you can't. Rearranging the words may change the meaning; I just want to know the correct way of rearranging them, although it would be interesting at some point to learn how those meanings change.
If you can say "Non era mia intenzione", then can you also say "Era mia intenzione"?
If that's the case, then "era mia casa" would also be OK - and also "era mia casa bianca", the point being that, after the verb, you drop the article. Unless there's something else I'm missing.