According to my boyfriend who's fluent, there's a really subtle difference. It takes a whole lot of time to discern the difference between them, and a whole lot more time to do it as second nature

I am still pretty new to this, but I think if they were saying "barna" then they would have to say "barna mine" instead of "barnet mitt". That may be one way to help differentiate between when it's "barna" and "barnet".

The placement of the possessive changes the form of the noun. Jeg elsker mitt barn is legal although not that widespread in use (it's more poetic/dramatic). The possessive is mitt as the gender of the noun barn is neuter. It is the same child anyway. But mitt barn could imply there is some other child you don't love.

mitt barn possessive + indefinite

barnet mitt definite + possessive

meg is not a possessive article, it's a personal pronoun, "me". Jeg - meg = "I - me"